apparently they are now using 12 volt only refrigerators. if you have one of these it would not need an inverter as it would be powered directly from the batteries. shore power does not run the refrigerator, though it will be needed at some point to recharge the batteries.
if you will post make / model of the refrigerator this can be confirmed.
2015 cardinal model 3825fl
2015 dodge ram 3500 dually