Q:

Did the LDS ever recognize non-LDS polygamous marriages? 

The Church of Jesus Christ of Latter-day Saints (LDS) at one time performed and recognized polygamous marriages.  In 1890, they ceased performing new polygamous marriages but continued to recognize polygamy (existing polygamous relationships were not dissolved, but no new polygamous marriages could be performed by the LDS church). 
Did the LDS church ever recognize, or have a procedure for recognizing, polygamous marriages entered into under the rules of another religion?  For example, Islam permits a man to have up to four wives.  If a polygamous Muslim converted to the LDS faith, was there (or is there) a structure, policy, practice, or principle in place to recognize them immediately as a polygamous Mormon upon conversion, or were polygamous marriages "stripped" from converts in a sense? 
Marriages performed outside of the LDS church are recognized by the LDS church as valid (though not eternal), but it is unclear if only monogamous marriages are thus recognized or whether they consider any marriage that is not inherently repugnant to the laws of God to be valid.  Islam does recognize non-Muslim polygamy for persons converting to Islam to the extent that the polygamy does not violate the rules of marriage in Islam (in the referenced hadith, a convert to Islam who already had more wives than is allowed in Islam was allowed to choose which wives to keep), so it stands to reason that accommodation of polygamous converts to a faith is a realistic concern. 
I will admit that mid-1800's Middle America wasn't exactly swarming with polygamous Muslims or polygamous followers of other faiths, so this was not a day-to-day concern, but still someone must have thought about the theory and what would eventually be done if the sphere of polygamous Mormons were to expand across the world. 
If the LDS church had a procedure for adjudicating the validity or acceptability of a polygamous marriage solemnized outside of the LDS faith, that counts as an answer, even if that procedure was never invoked in any actual cases. 

A:

The Church has never recognized polygamous unions formed outside the Church.   The authority to enter into a polygamous union is given in Doctrine & Covenants 132 and can only be done (when it was authorized) inside an LDS temple (when temples were available) by Priesthood holders authorized to use the Sealing Keys.   Hypothetically, back when the Church was authorizing polygomous unions, someone married polygamously outside the Church would be required to be sealed by proper authority, in the proper place.   Which is a long way of saying, no, we have not and do not recognize the authority of other religions to create polygamous unions. 
Having said that, it should be noted that the LDS church accepts marriages performed outside our church from a civil point of view, but we consider the marriages to be temporary, "until death do we part."   An LDS marriage (called a Sealing, "for time and all eternity") can only be performed by the same sealing authority I mentioned above in our temples.   In that regard, we don't recognize marriages performed by other religions at all.