[wisty]

tldr: They have to fly a flag, and follow the laws of whatever country's flag they are flying.

Under The United Nations Convention on the Law of the Sea, they would be in a US Exclusive Economic Zone - between 12 and 200 nautical miles of the coast, so no fishing or mining without US permission. As it's not Territorial waters (up to 12 nautical miles), US law will not apply, but there are international laws against Piracy in the High Seas which do. Of course, that only protects you from people who come from another boat.

As for what happens on the boat, it's governed by the laws of the country that they choose to be registered under. Let's say Panama. But they still have to follow some legal system. Panama law is based on Spanish traditions, and is probably acceptable.