TRUTH OR CONSEQUENCES, N.M.—A twin-fuselage jet took off on Sunday carrying a Virgin Galactic rocket plane primed to soar more than 50 miles above the New Mexico desert with British billionaire Richard Branson aboard in the vehicle’s first fully crewed test flight to space. Branson, one of six Virgin Galactic Holding Inc employees strapping in for the flight, has touted the mission as a precursor to a new era of space tourism, with the company he founded poised to begin commercial operations next year. Sunday’s high-altitude launch of the VSS Unity rocket plane marks the company’s 22nd test flight of its SpaceShipTwo system, and its fourth crewed mission beyond Earth’s atmosphere. It is also the first to carry a full complement of space travelers—two pilots and four “mission specialists,” Branson among them. A week away from his 71st birthday, Branson and his crewmates walked onto the tarmac of New Mexico’s …