Once you have found a tavern walk up to the bar and interact with the highlighted bottle. Each drink will cost 5R. You will need to have 5 drinks, each making Edward progressively drunker. Once you have downed the fifth, you will pass out and wake the next morning in either a haystack, a pile of leaves or a pile of banana fronds depending on the location of the tavern.
The first tavern I visited was on Nassau, pretty much right in the middle of the island.
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as confirmed by xK1, this does not work in the Kingston tavern. It seems it must have to be the regular open-air type taverns. so far, it has worked for me on both Nassau and Great Inagua.