Date: 202008
Local time of day: 1201-1800
Location: ZZZ.Tower, US
Flight conditions: VMC
Weather elements / visibility: 10
Light: Daylight
Aircraft: Gulfstream G280
Operator: Corporate
Operating under: Part 91
Flight plan: IFR
Mission: Passenger
Flight phase: Landing
Airspace: Class C ZZZ
Second aircraft: Texan T6/Harvard (Antique)
Reporter organization: Corporate
Reporter function: Pilot Flying; Captain
Reporter qualification: Flight Crew Instrument; Flight Crew Flight Instructor; Flight Crew Air Transport Pilot (ATP); Flight Crew Multiengine
Human factors: Situational Awareness; Confusion; Communication Breakdown
Second reporter function: First Officer; Pilot Not Flying
Anomaly: ATC Issue All Types; Conflict Ground Conflict; Critical; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy
Contributing factors: Human Factors; Procedure
Primary problem: Human Factors

Narrative: We were scheduled to arrive at ZZZ on a direct flight from ZZZ1 in Aircraft X. We were the first of at least three arriving aircraft making the transition from the en-route environment into the airport. We confirmed with approach control that we had visual contact with the airport and were cleared by ATC for the visual approach. Shortly thereafter; we were transferred to ZZZ Tower. Ahead of us was a flight of 2 Texan Navy trainers practicing maneuvers in the traffic pattern of the airport. While on a 4-mile final; we heard tower clear the flight of two Navy Texans to land in front of us. The lead Texan was perhaps several hundred feet above the runway and about to land. The second Texan was approximately at our altitude (perhaps a thousand feet above the runway) and perhaps a half mile behind the lead Texan. The lead Texan landed without incident. We slowed to our slowest possible airspeed to accommodate the traffic ahead and provide as much spacing as possible. The PM (Pilot Monitoring) mentioned to be prepared for a Go-Around. The Tower controller advised the second Texan to Go-Around and rejoin the downwind. Our aircraft was then given clearance to land. We maintained a slower airspeed so as to allow the lead Texan time to vacate the runway and so as to not create a runway incursion. Once the Texan had cleared the runway; we continued our descent. We landed without incident and had planned on exiting the runway with a left turn off onto taxi-way D which would allow for a shorter taxi to the FBO. As we slowed and passed the intersection of the runways we realized that the first Texan was stopped on the taxi-way (D) and did not have enough space behind him for us to vacate the runway with the left turn onto D. Further down the runway and off to the right; we noticed that a Coast Guard helicopter was waiting on taxi-way A5 to cross runway from A5 to C2. At this point; the PF (Pilot Flying) had come to a stop on the runway and was then planning a right turn out onto taxi-way A4 with the plan to circle back and use taxiway D to get to the FBO.While starting to move forward; Tower asked us which FBO (Fixed Base Operator) we were going to. The PM (Pilot Monitoring) replied that we were going to the FBO. The Tower Controller responded that they had two FBOs on the field. One being FBO1 and the other being FBO2. (Both of the FBOs are geographically located on the Southeast section of the airport; South of D and East of C). The PM responded that we would be going to FBO2. ATC stated that FBO2 had a lot of traffic on their ramp. He also mentioned that the FBO normally wanted traffic to enter via taxi-way C1; but due to the Texan blocking the left turn off available on D; we were to go ahead and continue down the runway and taxi to FBO2 via taxi-way C2.With the new clearance given; the PF applied power so as to not take the previously determined right turn off onto A4 and proceed forward toward the designated left turnoff onto C2. While abeam taxi-way A4; Tower called over the radio to never-mind using C2 and instead to make the hard-right turn and exit runway onto taxiway A4. The PM responded to the direction. Once he finished the statement; Tower told us to expedite off the runway. We complied as quickly and as safely to make the greater than 90-degree turn back toward A4.As we made the more than 90-degree turn; the crew noticed that a Navy Texan (presumably the second of the previously seen flight-of-two) had landed on runway XX; which we were still currently on. We had not cleared the runway at this point. No radio transmissions were heard or recalled by the crew about the Texan having been given clearance to land on runway XX nor did we hear that the Texan should execute another Go-Around. We exited the runway on A4; made the right turns onto taxiway A; and D and held short of runway XX as per the newly given direction by ATC. Tower said nothing either to the G280 and nothing to the Navy Texan. We continued viaour clearance to the FBO without further incident.Not knowing all of the allowable rules at that airport for military traffic in conjunction with civilian traffic; I think that perhaps the tower controller became fixated on getting our plane off the runway in the most expeditious manner possible. While the Texan was able to land in a distance allowing for virtually no 'separation issues'; we still were on the runway. Perhaps no rules were broken but this could easily have created an issue of non-compliance and resulted in serious injury or even death. While we were attempting to fulfill the requirements of the transmission; either the landing clearance for the second Texan was not given; or if given; not revoked since we were still on the runway. A corrective action on our part for future flights will be to brief the controller of our desired FBO prior to landing in an attempt to facilitate a more expeditious departure from the runway and to take the first available taxi-way off the runway unless expressly told otherwise by ATC.


Second narrative: [Report narrative contained no additional information.]


Synopsis: G280 flight crew encountered a ground conflict with another aircraft on the runway.