Other mistake: The scene in Klink's quarters is one of a few where you can see Larry Hovis' (Carter) wedding ring.
Continuity mistake: After Hogan, LeBeau, and Newkirk leave, Carter is sitting, holding the glass with both hands. After the cut, he's holding the glass in his right hand only.
Visible crew/equipment: After Hogan goes down into the tunnel in Klink's quarters, the stove moves back in place. As it does, the wire on the floor pulling the stove back in place is visible.
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.
raywest
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.