In Mafia III, there are two achievements or trophies related to alligators. Bon Appetit wants you to feed a body to the reptilian beasts, while Next Time Swim Faster simply asks for you to get eaten by them.
To get Bon Appetit, you’ll first have a kill an enemy near a large body of water that also contains alligators. Once you kill them, pick up their body and drop it in the water near one of those things. During the prologue where you have to kill the Haitian leader, Baka, you can accomplish this easily while sneaking around in the dead of night.
For Next Time Swim Faster, well, no more needs to be said about that. Just find the same body of water and look for an alligator, jump in, and wait for them to come around. It won’t take long for them to feast on your body, and as they chew you up, you’ll get the achievement or trophy. Keep in mind that it will take the gator about three bites to fully eat you up. Your trophy or achievement won’t pop until then.
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