First: Now I know who to ask if I have an electrical problem--FENZO!
Second: I'm thinking it has something to do with the hazard flasher system. I'm pretty sure you have to pull out the instrument panel to get to the flasher connector (what a PITA--as I have recently found out first-hand by changing to Speedhut gauges), but if you can or are in there for some reason, try pulling off the connector and see if that changes the results.
Third: Many German cars have an odd feature where if you leave the turn signal on, that tail light will stay illuminated even with the key off. It's supposed to be some kind of safety feature. I have no idea if this has anything to do with the FGT, but try switching that turn signal off and on with your key off and see if there's any correlation.
Second: I'm thinking it has something to do with the hazard flasher system. I'm pretty sure you have to pull out the instrument panel to get to the flasher connector (what a PITA--as I have recently found out first-hand by changing to Speedhut gauges), but if you can or are in there for some reason, try pulling off the connector and see if that changes the results.
Third: Many German cars have an odd feature where if you leave the turn signal on, that tail light will stay illuminated even with the key off. It's supposed to be some kind of safety feature. I have no idea if this has anything to do with the FGT, but try switching that turn signal off and on with your key off and see if there's any correlation.