I just started recording and I've got this really weird phaser sound when I try to record. I've tried both Reaper and Audacity but both end up with the same problem. The sound starts out fine but at about 5 seconds into recording audio quality decreases a lot and makes it sound very liquidy and phaser-like.

Does anyone know what the problem could be or any possible fixes?
Obviously Fassa.
Almost sounds like you're getting a echo that is slightly out-of-phase with the main signal. Are you sure you're not feeding your monitor signal back into the main?
I've got the microphone plugged into Mic 1 on my mixer and I have the rca cables connected directly from the mixer to the computers aux in port.
I'm new to this so bear with me if I'm too vague.
Obviously Fassa.
It would also be helpful if you could post an example recording.

It's probably not big news to you, but using your soundcard isn't always the best way to go. Most soundcards are fairly bad. My guess is it has something to do with the fact you're coming out of a Line Out on your mixer and going into a Mic In on your soundcard. More than likely, you have an impedance mismatch. It may even be overloading the Mic In on your soundcard.
It also does it if I play a few seconds, stop, then play again for a few seconds all on the same track. Does it still seem like a soundcard issue?
Obviously Fassa.
Soundcard/mixer/impedance issue. Did you try reducing the audio levels of the mixer and soundcard? I'd be curious to know if it still does it.
it could be monitor feedback, or a feedback loop somewhere in the signal flow
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I lowered the volume on the mixer and it kept doing it. How do I lower the soundcard levels? Is there some kind of adapter I can get to match the impedance or do I have to get something like a PODxt or an external soundcard?
Obviously Fassa.
Something like a PodXT is much better suited to recording than is your computer's soundcard. Prior to buying my Tascam US-2000, I also used a computer soundcard, but I wasn't impressed.