Propane/electric (gas absorbtion) fridges, running on electricity, use much more power than compressor type refrigerators - by about a factor of 5.  This is because they consist of a heat engine followed by a heat pump - all in one set of working fluids.

Running on propane they're pretty efficient.  Heat engine coupled to a heat pump.  Not as great as they COULD be but still moderately OK.

But running on electricity they use a resistance heater to turn the electrc power into heat.  That applies the carnot-cycle multiplier (about a factor of 1/3) plus the distribution losses to the efficiency of the heat engine/heat pump combo.  You burned a BUNCH of fuel at a generator plant to turn maybe a thrid of the energy into electricity (which you COULD have used at high efficiency to turn a compressor) and instead disipated it in a resistor to run ANOTHER heat engine.

Gas absorbtion fridges with an electric option are for situations where your primary power supply is a gas fuel (i.e. propane), but you sometimes can't run them on propane, have unmetered electric power handy, or where the electric price is outweighed by convenience (i.e. while towing a trailer, at trailer parks, when parked at home setting up for a trip.)

= = = =

Peltier cells are also inefficient - because after pumping heat across a gap using electricity (pretty efficiently) they leak most of it back again (because the thermal consuctivity of the semiconductors used is high).  Convenient if you only need a little cooling and/or have  power to burn (like in a car, where a kilowatt electric load steals less than one horsepower from the engine - which is burning more than that at idle just to keep from stalling).

There is a new tech coming that uses electrons across a vacuum (a REALLY  good thermal insulator) rather than electron/hole conduction in a semiconductor, to do peltier-style heat pumping / thermal generation with great efficiency.  But it isn't here yet.

= = = =

Bottom line, though, is that if you're trying to cool your food supply with an ambient electric power system (solar photovoltaic, wind, water), you'll use a factor of several less power with a compressor fridge than with a propane/electric OR a peltier.  Far more than enough to pay the inverter efficiency penalty to drive a regular appliance if you can't find one with a 12 volt compressor motor.

= = = =

Interestingly, I have a small (14" x 15" x 10 1/2") 120/12v compressor chest fridge that my wife got me a few years back.  "Norcold Tech II".  We use it on shopping trips and commutes between the houses (5 hours) to keep the food cold.  Very quiet.  Continuously adjustable from moderately cool through standard refrigerator to sub-zero freezer.  I never felt a need to open it up and see whether the compressor proper is run off an inverter for 12, has a 12 volt motor regulated down from 110, or what.

Norcold makes both gas absorbtion and compressor fridges and freezers (they're the largest maker of trailer fridges).