Factual error: In the rec hall where Klink is weightlifting, there is an advertisement poster for the Grindelwald region on the wall, which is in Switzerland. The German propaganda poured a lot of money into promoting the Reich's own recreational regions through the "Kraft durch Freude" program. With the war going badly, they especially wouldn't want a "Look how beautiful Switzerland is" poster in plain view of their troops every day.
Visible crew/equipment: As Klink and Schultz approach the barracks where Le Beau is "weightlifting", there is a cord or cable along the ground that would be for studio equipment.
Continuity mistake: When the woman is kissing Klink in town, she hands a note to the boy who puts the note inside the spare tire's hubcap. There was no tape to stick it anywhere, yet when LeBeau goes to get the note, it is stuck with tape to the center of the spare tire.
Answer: The core POWs regularly escaped the prison camp but they made it their mission to conduct espionage and commit sabotage in the surrounding German territory. They also collaborated with different underground resistance groups and used a network of secret tunnels to help prisoners from other POW camps to escape, who then relayed vital information back to the Allied forces. Hogan and his men maintained the illusion that Stalag 13 had never had any prisoners escape in order to avoid their covert operations being shut down. Being that the prison camp is set in Germany during WWII, there were no American embassies.
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Hogan has mentioned to different characters that they are actually stationed at Stalag 13 to help allied soldiers and prisoners from other Stalags to escape Germany.