Chinchorro is by far the hardest of the forts; not due to its defenses but due to the ships that come to its aid and the worsening storm that surrounds it. As you engage the fort with Mortars and cannonfire it will begin to rain and the surrounding ships will come to the forts aid. The longer you take, the worse the situation will become until you are defeated. To prevent defeat, you need to finish this quickly by trying to approach the fort without drawing the attention of any nearby frigates and then going all out as quickly as possible.
Once you are on the ground, enter the fort via the right side of the main gate and hop down to the hanging cages and cross them to reach the first Captain. Climb the walls and head for the second Captain, who you can Smoke Bomb and defeat, and then climb up again and approach the final Captain, who will run right to you and his death sentence. Once all Captains are defeated you can enter the war room and kill the forts Commander.
After the fort is cleared you can get the Treasure x2, Animus Fragment x2, and start the Naval Contract, Contraband.
