CAYE CAULKER, Belize — As I stepped inside the Barrier Reef bar, I heard the last thing I ever would have expected while on a tiny island off Belize: My name. “Amber!” yelled Mike the Bartender.

Caye Caulker is a sleepy backwater island nestled between the Belize coast and Belize Barrier Reef. With warm waters, cool breezes and friendly locals, it’s a little slice of Caribbean paradise ...