A new switch was the first thing I did!
Besides checking with a multi-meter, I've traced the wires from the switch to where they disappear into the dash with a huge bundle of other wires with no luck.
It IS possible for the lights to be on with no current in the harness: in spite of the physical switch being similar to old ones the circuit is different. In old style circuits, the reverse light switch controls the current that goes directly to the lights (as simple a circuit as you can imagine); in newer Bimmers it just sends a signal to one of the modules to tell the computer the vehicle is in reverse (so it tilts the passenger mirror etc). If there's a problem with the module, the wiring might be fine but the car still thinks its in reverse.