What happens to a country if it loses access to the world’s money?

On the Marshall Islands--a chain of volcanic islands in the middle of the Pacific Ocean, population 53,066—the only homegrown bank can’t issue credit cards, doesn’t have any ATMs and sends well-worn dollar bills between its island branches by boat. But it had been able to let islanders send and receive money from abroad through a link with Honolulu-based First Hawaiian Bank.

Now,...