ROME — Matthew Festing — aka His Most Eminent Highness The Prince and Grand Master of the Knights of Malta — bounds into the sitting room of his magnificent Renaissance palazzo sweaty and somewhat disheveled, and asks an aide if he should take off his sweater to be photographed.

Garrulous and self-effacing, Festing embodies some of the paradoxes of a fabled Catholic religious order that dates from the medieval Crusades: Steeped in European nobility and mystique, the order’s mission is humility and charity — running hospitals, ambulance services and old folks’ homes around the globe. It has many trappings of a country, printing its own stamps, coins, license plates and passports, and yet — a stateless state — it rules over no territory....