WASHINGTON—In a Pentagon hallway hung an austere portrait of a Navy man lost at sea in 1908, with his brass buttons, blue-knit uniform and what looks like meticulously blow-dried hair.

Wait. Blow-dried hair?

The portrait of "Ensign Chuck Hord," framed in the heavy gilt typical of government offices, may be the greatest—or perhaps only—prank in Pentagon art history. "Chuck Hord" can't be found in Navy records of the day. It isn't even a real painting. The textured, 30-year-old photo is actually of Capt. Eldridge Hord III, 53 years old, known to friends as "Tuck," a military retiree with a beer belly and graying hair who lives in Burke, Va.

Most military officers who climb the ranks or command daring battles only dream of having a portrait hang in a corridor of power at the Pentagon alongside the likes of Patton, Nimitz and Eisenhower. Capt. Hord's made its way to the Pentagon's C-ring hallway via several parties, an alliance of British and Canadian military officers and a clandestine, predawn operation later dubbed "THE PROJECT."...