I had the same problem about 3 years ago on my 2007 Cali, out of warranty, and spent a lot of time investigating batteries, fuses, and replacing the earth wire which was one possible source of this kind of problem with earlier models. This didn't result in a cure, so I googled further, including similar problems with marine fridges, which suffer similar problems, it seems.
I seem to remember that this site had a lot of discussion on fridge problems at the time, and that the recommended VW "cure" was a replacement fridge at about £1200!
With the help of the site, I eventually narrowed the possibilities down to a faulty fridge controller ( a small black box under the fridge, and hard to get to).
I bought a replacement on the Internet, and installed it, which luckily finally fixed the problem. It seems that the mechanical parts of the fridge, especially the compressor, are relatively "bullet-proof", and that the controller, with a printed circuit, is the weak link.
If your Cali is under warranty, I would expect VW to take care of it, without a big delay.
If your Cali is not in warranty, and VW are not willing to help, the same controller is used in a lot of fridges, including portable and marine types, and is made by Danfoss. It is described as a "Danfoss BD35F/BD50 CONTROLLER 12/24V". The (penguin) part number seems to be
E50895. The Danfoss part no. may be 101N0300
As compared with a possible bill for £1200, I thought a cost of about £100 for a new controller was worth a try.
I bought mine in 2013 from penguinfrigo.com, (Penguin Refrigeration Ltd, Unit 2B, Alexandria Park, Penner Road, Havant, Hants, PO9 1QY). They are still online. The site has some helpful advice on common fridge problems.
If you are into DIY and no longer worried about voiding your warranty etc ......
Installation involves removal of the fridge lid, unplugging of the power, etc. to the fridge through the small panel at the rear of the kitchen unit (beside the waste tank drain tap) the lifting out of the fridge and its compressor, etc underneath, (easier with two people) and removal of the black box and its replacement by the new one, a "plug in" job. The most difficult bit is getting everything past the cooling tubes and cage which surround the controller. I can't guarantee that this will solve all problems, as other causes may be involved. I have had no new fridge problems in the last 18 months.
I hope this is of some use.