Having your fridge wired correctly is very important. Some caravan 3-way fridges have their 12v connected directly to the caravan battery, however this will soon lead to a warm fridge, and a flat battery, they can consume a massive 15-25amps per hour. The solution to this is to rewire your 3-way caravan fridge to the tow vehicle, via the 12pin trailer plug, or its own Anderson plug.
Your caravan 3-way fridge should only operate on 12v while in transit (powered by the vehicle), and when you arrive at your destincation you should switch to 240v or gas. This setup keeps your auxillary battery at 100% when you arrive. Caravaners who do run their fridge off the caravan battery tend to have multiple batteries and a roof full of solar panels in order to keep up, or they have a 12v compressor fridge that can use less than 5 amps.
Portable fridges are an excellent investment, they can use as little as 1amp per hour and they will nearly always come with a plug ready for a 12V socket. However, often the factory sockets in the back of 4x4s are not up to scratch, constructed with cheap plastic and even cheaper wiring. This wiring cannot handle the constant current your fridge needs to stay cold, which could result in the socket being damaged or your fridge not working correctly. To keep the fridge cold, make sure you have a good connection and the right size wiring with circuit protection.