What is your 12V power source? If it's an old(er) converter, it may be putting out very unclean, spikey power--basically rectified AC rather than smooth DC--and a multimeter could be responding to the peak voltage, rather than the average or RMS or whatever.
While it's hard to diagnose exactly what's going on over a forum--goodness knows it's sometimes very hard when you're there in person--strange, inexplicable changes in voltages are often the result of bad ground connections, sometimes in places that are not immediately obvious. Check them all carefully. I'd pay particular attention to the connections at the battery, and also verify that the battery cables themselves haven't corroded inside the insulation, which I've seen happen a couple of times. The terminal is tight on the battery, but the wire just past it is mostly (or entirely) powdered crud inside the insulation jacket.