<PAGE>   1
 
================================================================================
 
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

                             ---------------------
 
                                   FORM 10-Q
 
[X]             QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d)
                     OF THE SECURITIES EXCHANGE ACT OF 1934
                 FOR THE QUARTERLY PERIOD ENDED MARCH 31, 1999
                                       OR
[ ]            TRANSITION REPORT PURSUANT TO SECTION 13 OR 15(d)
                     OF THE SECURITIES EXCHANGE ACT OF 1934
 
             FOR THE TRANSITION PERIOD FROM ________ TO ________ .
                         COMMISSION FILE NUMBER 0-8565
 
                             MARINE PETROLEUM TRUST
             (Exact name of registrant as specified in its charter)
 
                    TEXAS                                        75-6008017
         (State or other jurisdiction                         (I.R.S. Employer
      of incorporation or organization)                     Identification No.)

 NATIONSBANK, N.A., DBA BANK OF AMERICA, N.A.                    75283-0241
        P.O. BOX 830241, DALLAS, TEXAS                           (Zip Code)
   (Address of principal executive offices)
 
       Registrant's telephone number, including area code (800) 985-0794
 
                                      None
              (Former name, former address and former fiscal year
                         if changed since last report)

                             ---------------------
 
     Indicate by check mark whether the registrant (1) has filed all reports
required to be filed by Section 13 or 15(d) of the Securities Exchange Act of
1934 during the preceding 12 months (or for such shorter period that the
registrant was required to file such reports), and (2) has been subject to such
filing requirements for the past 90 days.
 
                             Yes  X       No
                                 ---         ---

     Indicate number of units of beneficial interest outstanding as of the last
practicable date.
 
Title of Each Class of Units              Number of Units of Beneficial Interest
   of Beneficial Interest                             March 31, 1999
----------------------------              --------------------------------------

UNITS OF BENEFICIAL INTEREST                            2,000,000
 
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<PAGE>   2
 
                             MARINE PETROLEUM TRUST
 
                                     INDEX
 
<TABLE>
<CAPTION>
                                                               PAGE
                                                              NUMBER
                                                              ------
<S>                                                           <C>
                PART I. FINANCIAL INFORMATION
 
ITEM 1. FINANCIAL STATEMENTS (UNAUDITED)....................     2
 
Condensed Consolidated Balance Sheets March 31, 1999 and
  June 30, 1998.............................................     2
 
Condensed Consolidated Statements of Income and
  Undistributed Income for the Three Months and Nine Months
  Ended March 31, 1999 and 1998.............................     3
 
Condensed Consolidated Statements of Cash Flows for the Nine
  Months Ended March 31, 1999 and 1998......................     4
 
Notes to Condensed Consolidated Financial Statements........     5
 
ITEM 2. MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL
        CONDITION AND RESULTS OF OPERATIONS.................     6
 
ITEM 3. QUANTITATIVE AND QUALITATIVE DISCLOSURE ABOUT MARKET
        RISK................................................     9
 
                   PART II. OTHER INFORMATION
 
ITEM 6. EXHIBITS AND REPORTS ON FORM 8-K....................     9
 
Signatures..................................................    10
 
Exhibit Index...............................................    11
</TABLE>
<PAGE>   3
 
                         PART I. FINANCIAL INFORMATION
 
ITEM 1. FINANCIAL STATEMENTS
 
                     MARINE PETROLEUM TRUST AND SUBSIDIARY
 
                     CONDENSED CONSOLIDATED BALANCE SHEETS
                        MARCH 31, 1999 AND JUNE 30, 1998
                                  (UNAUDITED)
 
                                     ASSETS
 
<TABLE>
<CAPTION>
                                                               MARCH 31      JUNE 30
                                                                 1999          1998
                                                              ----------    ----------
<S>                                                           <C>           <C>
Current Assets:
  Cash and cash equivalents.................................  $1,841,135    $1,922,336
  Oil and gas royalties receivable..........................     290,099       399,007
  Receivable from affiliate.................................      91,422       116,377
                                                              ----------    ----------
          Total current assets..............................   2,222,656     2,437,720
                                                              ----------    ----------
Investment in affiliate.....................................     332,945       306,670
Office equipment, at cost less accumulated depreciation.....         555           555
Producing oil and gas properties............................           7             7
                                                              ----------    ----------
                                                              $2,556,163    $2,744,952
                                                              ==========    ==========
 
                             LIABILITIES AND TRUST EQUITY
 
Current Liabilities:
  Accounts payable..........................................  $  452,831    $  926,663
  Income taxes payable......................................       8,821         9,021
                                                              ----------    ----------
          Total current liabilities.........................     461,652       935,684
Trust Equity:
  Corpus -- authorized 2,000,000 units of beneficial
     interest, issued 2,000,000 units at nominal value......           8             8
  Undistributed income......................................   2,094,503     1,809,260
                                                              ----------    ----------
          Total trust equity................................   2,094,511     1,809,268
                                                              ----------    ----------
                                                              $2,556,163    $2,744,952
                                                              ==========    ==========
</TABLE>
 
                                        2
<PAGE>   4
 
                     MARINE PETROLEUM TRUST AND SUBSIDIARY
 
      CONDENSED CONSOLIDATED STATEMENTS OF INCOME AND UNDISTRIBUTED INCOME
       FOR THE THREE MONTHS AND NINE MONTHS ENDED MARCH 31, 1999 AND 1998
                                  (UNAUDITED)
 
<TABLE>
<CAPTION>
                                                    THREE MONTHS               NINE MONTHS
                                               -----------------------   -----------------------
                                                  1999         1998         1999         1998
                                               ----------   ----------   ----------   ----------
<S>                                            <C>          <C>          <C>          <C>
Income:
  Oil and gas royalties......................  $  648,852   $  562,845   $2,161,149   $1,996,217
  Equity in earnings of affiliate............     117,859      134,457      267,766      384,689
  Interest income............................      23,506       26,937       74,970       74,837
                                               ----------   ----------   ----------   ----------
                                                  790,217      724,239    2,503,885    2,455,743
                                               ----------   ----------   ----------   ----------
Expenses -- General and administrative.......      39,260       38,230      141,226      111,070
                                               ----------   ----------   ----------   ----------
          Income before federal income
         taxes...............................     750,957      686,009    2,362,659    2,344,673
Federal income taxes of subsidiary...........       3,100          400        4,200        6,100
          Net income.........................     747,857      685,609    2,358,459    2,338,573
Undistributed income at beginning of
  period.....................................   1,956,720    2,122,265    1,809,260    2,044,795
                                               ----------   ----------   ----------   ----------
                                                2,704,577    2,807,874    4,167,719    4,383,368
Distributions to unitholders.................     610,074    1,021,216    2,073,216    2,596,710
                                               ----------   ----------   ----------   ----------
Undistributed income at end of period........  $2,094,503   $1,786,658   $2,094,503   $1,786,658
                                               ==========   ==========   ==========   ==========
Net income per unit..........................  $     0.37   $     0.34   $     1.18   $     1.17
                                               ==========   ==========   ==========   ==========
Distributions per unit.......................  $     0.31   $     0.51   $     1.04   $     1.30
                                               ==========   ==========   ==========   ==========
</TABLE>
 
                                        3
<PAGE>   5
 
                     MARINE PETROLEUM TRUST AND SUBSIDIARY
 
                CONDENSED CONSOLIDATED STATEMENTS OF CASH FLOWS
               FOR THE NINE MONTHS ENDED MARCH 31, 1999 AND 1998
                                  (UNAUDITED)
 
<TABLE>
<CAPTION>
                                                                 1999           1998
                                                              -----------    -----------
<S>                                                           <C>            <C>
Cash flows from operating activities:
  Net income................................................  $ 2,358,459    $ 2,338,573
  Adjustments to reconcile net income to net cash provided
     by operating activities:
     Depreciation...........................................           --          1,936
     Equity in undistributed earnings of affiliate..........      (26,275)       (52,321)
     Change in assets and liabilities:
       Oil and gas royalties receivable.....................      108,908        570,755
       Receivable from affiliate............................       24,955        (45,759)
       Accounts payable.....................................     (473,832)       114,526
       Income taxes payable.................................         (200)           100
                                                              -----------    -----------
          Net cash provided by operating activities.........    1,992,015      2,927,810
                                                              -----------    -----------
Cash flows used in investing activities --
       Purchase of office equipment.........................           --         (2,219)
Cash flows from financing activities -- distributions to
  unitholders...............................................   (2,073,216)    (2,596,710)
                                                              -----------    -----------
          Net increase (decrease) in cash and cash
             equivalents....................................      (81,201)       328,881
                                                              -----------    -----------
Cash and cash equivalents at beginning of period............    1,922,336      1,759,718
                                                              -----------    -----------
Cash and cash equivalents at end of period..................  $ 1,841,135    $ 2,088,599
                                                              ===========    ===========
</TABLE>
 
                                        4
<PAGE>   6
 
                     MARINE PETROLEUM TRUST AND SUBSIDIARY
 
              NOTES TO CONDENSED CONSOLIDATED FINANCIAL STATEMENTS
                                 MARCH 31, 1999
                                  (UNAUDITED)
 
ACCOUNTING POLICIES
 
     The financial statements include the financial statements of Marine
Petroleum Trust (the "Trust") and its wholly-owned subsidiary, are condensed,
and should be read in conjunction with the Trust's annual report on Form 10-K
for the fiscal year ended June 30, 1998. The financial statements included
herein are unaudited, but in the opinion of management include all adjustments
necessary for a fair presentation of the results of operations for the periods
indicated.
 
UNDISTRIBUTED INCOME
 
     Undistributed income on March 31, 1999 includes $1,320,779 applicable to
the Trust and $773,724 applicable to Marine Petroleum Corporation, the Trust's
wholly-owned subsidiary. Distributions to unitholders are dependent on the
volume and price of oil and gas sold by others and will fluctuate from quarter
to quarter.
 
ACCOUNTS PAYABLE
 
     Marine Petroleum Corporation has provided an account payable of $452,250 to
cover possible refunds that may be required upon redetermination of gas prices
for royalty payments in prior periods. During the current quarter, Marine
Petroleum Corporation determined that $102,000 of the previously established
account payable was no longer required. Accordingly accounts payable was reduced
and oil and gas royalties was increased by $102,000.
 
                                        5
<PAGE>   7
 
ITEM 2. MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND RESULTS
OF OPERATIONS
 
FINANCIAL CONDITION -- LIQUIDITY AND CAPITAL RESOURCES
 
     The Trust is a "royalty trust" with overriding royalty interests in oil and
gas leases in the Gulf of Mexico. The Trust's indenture (and the charter and
by-laws of its subsidiary) expressly prohibit the operation of any kind of trade
or business. All royalties received by the Trust, less administrative expenses,
are distributed quarterly to unitholders. Since the Trust's sole purpose is to
collect and distribute cash collected from royalties, there are no requirements
for capital.
 
GENERAL
 
     Net income for the three months ended March 31, 1999 amounted to $.37 per
unit which was up approximately 9% from the $.34 realized for the three months
ended March 31, 1998. Income from royalties increased approximately 15%,
including the additional royalty income referred to in the "Notes to Condensed
Consolidated Financial Statements -- Accounts Payable." The Trust's equity in
the income of Tidelands Royalty Trust B decreased approximately 12% for the
current three months as compared to the comparable period a year ago.
 
     Distributions fluctuate from quarter to quarter. The amount of each
distribution is determined by the amount of cash available for distribution on
the date the distribution is determined. The amount of each distribution is
determined 10 days before the applicable record date.
 
     The following table presents the net production quantities of oil and gas
and net income and distributions per unit for the last five quarters.
 
<TABLE>
<CAPTION>
                                                                PRODUCTION
                                                          ----------------------    NET         CASH
QUARTER                                                   OIL(BBLS)    GAS(MCF)    INCOME   DISTRIBUTION
-------                                                   ----------   ---------   ------   ------------
<S>                                                       <C>          <C>         <C>      <C>
March 31, 1998..........................................    10,301      175,595     $.34        $.51
June 30, 1998...........................................    15,422      153,557      .32         .31
September 30, 1998......................................    12,844      223,920      .34         .33
December 31, 1998.......................................    16,382      346,495      .46         .41
March 31, 1999..........................................    17,496      279,145      .37         .31
</TABLE>
 
     The Trust's current fiscal year began on July 1, 1998. Since that date,
operators have completed 21 new development wells on leases in which the Trust
has an interest. Two of the wells were not successful while 19 were completed as
either oil or gas producers. At the present time operators have 9 development
wells in process and have identified 10 locations for development wells to be
drilled in the future. In addition to drilling operations on new wells, the
operators work over old wells by re-completing them in other producing
formations. Because operators do not report their drilling operations to the
Trust, the Trust must rely on public records for information regarding drilling
operations.
 
     The Trust's revenues are derived from the oil and gas production activities
of unrelated parties. The Trust's revenues and distributions fluctuate from
period to period based upon factors beyond the Trust's control, including
without limitation the number of productive wells drilled and maintained on
leases subject to the Trust's interest, the level of production over time from
such wells and the prices at which the oil and gas from such wells is sold. The
Trust believes that it will continue to have revenues sufficient to permit
 
                                        6
<PAGE>   8
 
distributions to be made to unitholders for the foreseeable future, although no
assurance can be made regarding the amounts thereof. The foregoing sentence is a
forward-looking statement. Factors that might cause actual results to differ
from expected results include reductions in prices or demand for oil and gas,
which might then lead to decreased production; reductions in production due to
depletion of existing wells or disruptions in service, including as the result
of storm damage to production facilities, blowouts or other production
accidents, and geological changes such as cratering of productive formations;
expiration or release of leases subject to the Trust's interests; and the
discontinuation by parties subject to the contract dated April 30, 1951 between
Tidelands' predecessors and Gulf Oil Corporation of their efforts to obtain
leases in the area that is subject to Tidelands' interests.
 
     Important aspects of the Trust's operations are conducted by third parties.
These include the production and sale of oil and gas and the calculation and
payment of royalties to the Trust, which are conducted by oil and gas companies
that lease tracts subject to the Trust's interests. Similarly, the Trust's
distributions are processed and paid by the Bank of New York. Any disruption of
the Trust's operations that results from Year 2000 problems of these third
parties could have a material adverse effect on the Trust. The Trust does not
have access to information that would permit it to determine the status of
certain of these third parties' efforts to analyze and address Year 2000 issues.
The Trust is monitoring these parties' public announcements and disclosure
regarding Year 2000 issues, and may seek direct assurances where appropriate, in
an effort to ensure that these parties address Year 2000 issues that could
adversely affect the Trust.
 
RESULTS OF OPERATIONS -- THREE MONTHS ENDED MARCH 31, 1999
 
     Net income for the quarters ended March 31, 1999 and March 31, 1998
amounted to $747,857 (including the $102,000 release of the accounts payable
reserve) which was approximately 9% more than the $685,609 (including the
$52,474 release of the accounts payable reserve) realized in the comparable
period of 1998.
 
     Revenue from oil royalties increased 25% to $149,331 in the current quarter
from $119,592 realized in the comparable period a year ago. As shown in the
table below, the increase in oil production over the comparable period a year
ago was offset by a decline in oil prices realized.
 
     Revenue from gas royalties increased 13% to $499,521 in the current quarter
from $443,254 in the comparable period a year ago. The gas revenue figures
include the additional revenue from the reduction of the accounts payable
reserve as discussed above. As shown in the table below, while gas production
increased over the comparable period a year ago, it was offset by a decline in
natural gas prices.
 
     Income from the equity interest in Tidelands decreased approximately 24% in
the current period over the comparable period last year due to a decline in oil
and natural gas prices partially offset by an increase in oil and natural gas
production.
 
                                        7
<PAGE>   9
 
     The quantities of oil and gas sold and the average prices realized (from
current operations without the revenues realized from the reduction of the
accounts payable reserve as described above and including the equity in
Tidelands) for the three months ended March 31, 1999, and those realized in the
comparable 1998 quarter, are presented in the following table:
 
<TABLE>
<CAPTION>
                                                                1999        1998
                                                              --------    --------
<S>                                                           <C>         <C>
OIL
  Barrels sold..............................................    17,496      10,301
  Average price.............................................  $   8.86    $  12.29
 
NATURAL GAS
  Mcf sold..................................................   279,145     175,595
  Average price.............................................  $   1.79    $   2.92
</TABLE>
 
RESULTS OF OPERATIONS -- NINE MONTHS ENDED MARCH 31, 1999
 
     Net income for the nine months ended March 31, 1999 amounted to $2,358,459,
(including the $306,000 release of the accounts payable reserve) which was
approximately 1% more than the $2,338,573 (including the $52,474 release of the
accounts payable reserve) realized in the comparable period of 1998.
 
     Revenue from oil royalties decreased 31% to $514,895 in the current nine
month period from $751,392 realized in the comparable period a year ago. As
shown in the table below, there was a decrease in oil production and a decline
in oil prices.
 
     Revenue from gas royalties increased 32% to $1,646,254 in the current nine
month period from $1,244,825 realized in the comparable period a year ago. As
shown in the table below, there was an increase in gas production over the
comparable period a year ago and a decline in natural gas prices.
 
     Income from the equity interest in Tidelands decreased approximately 30% in
the current period over the comparable period last year due to a decline in oil
and natural gas prices and in oil and natural gas production.
 
     Marine Petroleum Corporation has entered into an agreement with NationsBank
to provide collection, accounting and reporting services with respect to its
overriding royalty properties. During the current period general and
administrative expense was increased by a one time set up fee of $19,000.
 
     The quantities of oil and gas sold and the average prices realized (from
current operations without the revenues realized from the reduction of the
accounts payable reserve as described above and including the equity in
Tidelands) for the nine months ended March 31, 1999 and 1998, are presented in
the following table:
 
<TABLE>
<CAPTION>
                                                                1999        1998
                                                              --------    --------
<S>                                                           <C>         <C>
OIL
  Barrels sold..............................................    46,721      47,166
  Average price.............................................  $  13.61    $  16.92
 
NATURAL GAS
  Mcf sold..................................................   849,560     600,586
  Average price.............................................  $   1.91    $   2.14
</TABLE>
 
                                        8
<PAGE>   10
 
ITEM 3. QUANTITATIVE AND QUALITATIVE DISCLOSURE ABOUT MARKET RISK
 
     Not applicable.
 
                           PART II. OTHER INFORMATION
 
ITEM 6. EXHIBITS AND REPORTS ON FORM 8-K
 
     (a) The following exhibits are included herein:
 
<TABLE>
<CAPTION>
        EXHIBIT
         NUMBER                             DESCRIPTION OF EXHIBIT
        -------                             ----------------------
<C>                      <S>
          27.1           -- Financial Data Schedule
</TABLE>
 
     (b) Current Reports on Form 8-K:
 
          None
 
                                        9
<PAGE>   11
 
                                   SIGNATURES
 
     Pursuant to the requirements of the Securities Exchange Act of 1934, the
registrant has duly caused this report to be signed on its behalf by the
undersigned, thereunto duly authorized.
 
                                            MARINE PETROLEUM TRUST
 
                                            NationsBank, N.A. DBA Bank of
                                            America, N.A., Trustee
 
May 13, 1999                                By:      /s/ JANE J. SHEA
                                              ----------------------------------
                                                         Jane J. Shea
                                                        Vice President
 
May 13, 1999                                         /s/ R. RAY BELL
                                            ------------------------------------
                                                Principal Accounting Officer
 
                                       10
<PAGE>   12
 
                               INDEX TO EXHIBITS
 
<TABLE>
<CAPTION>
        EXHIBIT
         NUMBER                             DESCRIPTION OF EXHIBIT
        -------                             ----------------------
<C>                      <S>
          27.1           -- Financial Data Schedule
</TABLE>