I think I might've figured it out. I have a 2N5089 in the spot labeled MPSA18. Is that the definitive source of the distortion, or is MPSA18 also a transistor?
The 2N5089 and MPSA18 are both general purpose NPN transistors, with virtually identical 'on' and 'off' specs.
Additionally, those transistors control the compression ('Ratio') for LDR1, which acts basically like a volume pot which senses the AC signal intensity and leaks the signal to ground. It does not pass signal through to output. So, it is unlikely the cause of your distortion.
Other than the voltages I previously discussed; the other causes could be:
1.) The Ratio pot not being soldered well or not the correct value.
2.) The Sustain pot not being soldered well or not the correct value.
3.) Incorrect C6 or C20 values (from the picture, these look good).
The way this compressor works is:
IC 1.1 amplifies the signal according to the ratio and sustain settings (the more of each, the more amplification.
The more IC 1.1 amplifies, the more Q1 & Q2 sends current to the LED inside of LDR1, making that LED brighter.
Which causes the photoresistor inside of LDR of lose resistance and sends additional negative feedback to the IC 1.1
Which additional negative feedback lowers the amplification gain of the IC 1.1 circuit (same results as turning the Ratio down).
As you can see, this arrangement is constantly battling itself according to the amplification of the AC signal. As others have indicated, extreme clockwise settings of the Ratio and Sustain controls can cause distortion, as that 'battle' gets more intense. That battle can be adjusted by using different values for C6 and C20 (smaller cap values will make it less sensitive and larger ones will make it worse).
Distortion can also happen if Vref is not half of Vcc, as part of the signal waveform will get cut off when IC 1.1 amplifies. Ditto if the IC is shorted.
Note that IC 1.2 acts as an output buffer (unity gain - signal in = signal out) when the 'Level' is off (counterclockwise) and a small amplification as level is increased. The only way that can be creating the distortion would be Vref or a shorted IC.
Hope this helps.