Reading 25 VAC on the output of a 12 VDC power supply is definitely wrong. Unfortunately, from what you tell us it's hard to determine what exactly is wrong. Perhaps that supply is just broken.
The best thing to do would be to look at its output voltage on a scope. Then you can see for sure what is going on. There are other ways to get some idea about a AC signal. For example, put a speaker in series with a 1 kΩ resistor on the supply output. If it really has such large AC and it is in the audible range, then you'll definitely hear it. If it is really putting out 25 VAC RMS (which I have a hard time believing), then a 1 kΩ resistor will dissipate over 600 mW, which will make a ordinary "1/4 W" resistor get very hot quickly. If the voltage is really that large, you'll hear something with a 10 kΩ resistor in series with the speaker too.
You can also try putting some capacitance on the supply output and see what that does to the meter reading. To be safe, get capacitor rated for 50 V at least. You probably need 10s of µF before anything much happens. If this supply is truly broken, it could blow up the cap though. Again, a scope would tell us what's really going on.
I just had another thought as to what is going on. 25 VAC from a 12 VDC supplies seems a bit unbelievable, even for a busted one. I'm guessing your meter isn't really connectect accross the supply output properly. Does this supply possibly have 3 terminals? I have seen some where it's a bit confusing which two are actually the supply output and that the third is the wall plug ground. There is usually a strap you can clamp between the wall ground and one of the two supply ends. When this is not strapped and you put the meter between either output and the wall ground, you can get exactly what you are seeing. There will be some capacitance in the supply to the hot side of the AC line, and this will add a common mode signal onto the supply output. It is high impedance, so not really a problem. If you put a 10 kΩ resistor accross the meter when reading the AC voltage and it drops down a lot, then that's what's happening.
From your latest experiments, it sounds like the DC blocking cap isn't in series when you are doing the AC tests. Look closely at your meter. Does it only have two places to plug the leads into, or are there two or more jacks for the red lead depending on what you are trying to measure? When taking AC measurements, not only make sure the dial is on AC volts, but also that the leads are plugged into the correct places for AC voltage measurement.