Impossible Trinity is unlocked after completing the mission Museum Piece. This achievement recieved its name due to three of the Grand Theft Auto IV era protagonists appearing in this game, Niko Bellic from Grand Theft Auto IV, Johnny Klebitz from The Lost and Damned and Luis Fernando Lopez from The Ballad of Gay Tony.
 Protagonist revealed
This mission hinted at several websites and message boards about who the protagonist for The Ballad of Gay Tony would be. The prominent hint was that the third protagonist appeared in Museum Piece alongside Niko and Johnny, pointing to Luis Fernando Lopez, who was also seen at the end of the Lost and Damned, during the credits. 
 Other Appearances
The Ballad of Gay Tony has its own similar achievement, Diamonds Forever with the same achievement image and the mission description of "Complete the Trinity". It's unlocked after completing Not So Fast for Tony Prince.