The relatively small, flat island of Anguilla appeared as a scrub-covered outcropping on the horizon, an easily overlooked smudge on the surface of the sea.

Anguilla: With no cruise ships, casinos or high-rise hotels, this bohemian-upscale island sports some of the region’s most pristine beaches.

“Specifically, I said to Governor Christie that if you get elected president, I wanted to lay claim to the ambassadorship of Anguilla,” Mr. Wildstein said.