PARIS—In 1860, a French lawyer named Orélie Antoine de Tounens ventured into the wilds of southern Chile in search of glory. He met up with the fierce Mapuche Indians and somehow persuaded tribal leaders to make him king of a vast new Indian nation, the Kingdom of Araucania and Patagonia.

Just over a year later, the man calling himself King Orélie Antoine I was captured by Chilean troops, declared insane and booted off the continent.

Back in France, Mr. de Tounens persisted in claiming a South American realm, even after he was reduced to working as a lamplighter and living with his nephew, a butcher. The kingdom was never recognized by any other nation and its territory remains part of Chile and Argentina. But the royal house—supported by nobles, councilors and cabinet members—carries on to this day, even though it no longer asserts any territorial claims. Court members meet periodically to celebrate the élan of the first king, promote Indian rights—and, not least, award each other some really impressive-seeming medals and titles.

Now a bitter battle has broken out for leadership of a kingdom that only exists in history books. In January of 2014 the kingdom’s leader, Philippe Boiry, died after a 62-year reign as Prince Philippe, opening the door to a strange game of thrones.

A special council moved swiftly to elevate Jean-Michel Parasiliti di Para, a 73-year-old confidante of Mr. Boiry, to rule as Prince Antoine IV. The council said the election was held strictly according to the rules.