The force of Mr. Pierce’s personality and his spiritual presence are important to the group, whose members are otherwise largely agnostic. Mr. Pierce regularly performs rituals. Earlier that day while scoping out property, they had stopped at a historic Ceiba tree, known as the Tree of Life.

“Brock nestled into the bosom of it and was there for 10 minutes,” Mr. Nygard said.

Mr. Pierce walked around the tree and said prayers for Puertopia, holding a rusted wrench he had picked up in the territory. He kissed an old man’s feet. He blessed a crystal in the water, as they all watched. He played the Chaplin speech to everyone and to the tree, Mr. Nygard said.

That wrench is now in the penthouse, heavy and greasy.

Later on, at a dinner in a nearby restaurant, the group ordered platters of octopus arms, fried cheese, ceviche and rum cocktails. They began debating whether to buy Puerto Rico’s Roosevelt Roads Naval Station, which measures 9,000 acres and has two deepwater ports and an adjacent airport. The only hitch: It’s a Superfund cleanup site.

Mr. Pierce had fallen asleep by then, his hat tilted down and arms crossed. He gets two hours of sleep many nights, often on a firm grounding mat to stay in contact with the earth’s electric energy. Josh Boles, a tall, athletic man who is another crypto expat, picked him up, and the group headed back to the Monastery.

They walked past a big pink building in an old town square, the start of their vision for Puertopia’s downtown. Once a children’s museum, they plan on making it a crypto clubhouse and outreach center that will have the mission “to bring together Puerto Ricans with Puertopians.”

The Vanderbilt

Workdays are casual in Puertopia. One morning, Bryan Larkin, 39, and Reeve Collins, 42, were working at another old hotel, the Condado Vanderbilt, where they had their laptops on a pool bar with frozen piña coladas on tap.

“We’re going to make this crypto land,” Mr. Larkin said.

Mr. Larkin has mined about $2 billion in Bitcoin and is the chief technology officer of Blockchain Industries, a publicly traded company based in Puerto Rico.