LOS ANGELES—At the after-afterparty for Tuesday’s premiere of his war movie “Midway,” director Roland Emmerich posed for two photos back-to-back as he navigated the sushi tables and tiki torches around his home’s tile-lined pool.

One was with “Glee” star Darren Criss, who appears in the film as an American pilot fighting the Japanese in the Pacific. The other was with several executives from Starlight Culture Entertainment Group, a Chinese company that financed Mr. Emmerich’s movie.

“You’re...