en voulant faire une surprise à sa femme, il découvre qu'elle le trompe

In the Marines, a "Jody" is a generalized term meaning: any man who stays home while everyone else goes to war. He gets to enjoy all the things the Marines are missing, more specifically the Marine's girlfriend back at home while the Marine is away on active duty. The reason that they're called Jody specifically dates back to black soldiers in WWII. They took a character from old blues songs named Joe the Grinder (or Joe D. Grinder) who would steal the ladies of inmates and soldiers, and clipped his name to Jody.