Strange problem.

Whenever I connect to 110v hook-up and try to run
my fridge in electric mode, the fridge works fine -- 
for a while.  After abrief time, the lamp goes off 
and the fridge simply stops cooling.  The hook-up
seems to keep the RV battery charged just fine -- 
but the fridge will not run off 12v.

If that is not strange enough, I can disconnect from 
the 110v hook-up and the fridge will still not run 
off the battery!  Only when I start the engine and
rev the engine to the point that the alternator light
on the dash goes off will the Fridge "on" lamp come
back on!

I suspect crossed wires, a relay somewhere, etc., but
where to start?