The coin itself features Mr. Trump’s face embossed in gold on a base of shining silver. His slogan, “Keep America Great,” is written in an arc atop his head.

It could be yours, too, for just $0. (Plus $9.99 shipping and handling.)

This particular version has become something of an internet mystery. It’s not obvious who’s promoting it or profiting from it. It’s sold by a mysterious news website about which little is disclosed. Mr. Trump has nothing to do with the coin. No identifiable company is taking credit.

Some internet sleuths have followed digital bread crumbs to suss out the organizations behind the coin’s popularity. The New York Times took a similar path, tracking a network of advertisers and following their activity from public storefronts to private marketing portals. Using guidance from insiders and marketing veterans, the search crisscrossed the globe, implicating a handful of companies and a legion of anonymous marketers operating in parts unknown.