might be worth checking if there is 12v on it all the time or just when its in reverse?
an extension to that would be to check if there is 12v coming off the ground too.
a good example of 12v on 12v is the internal boot light for whatever reason mine turns off not by cutting the 12v line, but by pumping 12v down the ground. Didn't realise until I wired in a second boot light and grounded it off something else and it wouldn't turn off, had to run a second cable around to it to do the job
Anyhow, if its perhaps shorting out or damaged somewhere it might help explain it.
Could be that there is permanent 12v on one side of the bulb and the other terminal is grounded through the reverse switch.
Don't know for sure but it would account for what your seeing.
If this is the case then OP has a faulty reverse switch?