The Informer - S1-E1
Continuity mistake: The ground is soft, dark gray dirt. From the next episode on it is hard packed yellowish clay.
The Informer - S1-E1
Visible crew/equipment: As Burkhalter looks down at the floor under the water tower, the studio lighting positions are distinguishable on the visor of his cap.
The Informer - S1-E1
Continuity mistake: In this episode, the doghouse that disguises an entrance to the underground is flipped up on end, and the separate floor is flipped up in the opposite direction. In all of the following episodes, the doghouse, and the floor, are one piece.
The Informer - S1-E1
Continuity mistake: In this episode, Burkhalter holds the rank of Colonel, Carter is a Lieutenant. In subsequent episodes, Burkhalter is a General, and Carter is a Sergeant.
The Informer - S1-E1
Other mistake: Klink has on his monocle, as usual. If you look closely, the monocle is a single lens, with no frame. From Episode 2 onward, the monocle has a frame and a gallery. Also, Klink wears it at an angle, not unlike the Penguin in the original Batman series, who also wore a monocle without a frame.
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.