phase issues are probably the most overlooked source for any signal degradation - be it mix or processing.
Phase fidelity in the analog world makes the difference between high end and just good gear - it's not a digital phenomenon at all.
The effect is frequency = time depended and may of course influence any signal processing.
Often the sound gets a dull character - it's just not right without an obvious reason - but it can also get quite punchy if the signals match in a certain way.
I've noticed it impressively on 2 occasions.
Once when accidently monitoring a signal on the mixer and Pulsar. The adat connection makes a 0.5 ms delay and when I moved the fader up (with was a pad sound) there was a nice comb filter effect

I really didn't get it. There was no filter in the chain but it clearly sounded as if.
Another time when listening to a classic concerto with cheap speakers, when the solo violin suddenly panned from left to right. They really don't do these tricks at Sony...
My personal guess is that phase fidelity is also an important aspect of the various audio engines in sequencer applications, not only bit depth and samplerate.
But no need to get paranoid: any audio gear sufferes more or less from this.
cheers, Tom