The PA mix is meant to augment the live sound of the band and monitors, not to sound good by itself. Mike each instrument and singer. You can reduce leakage by miking close and recording direct. Have vocalists sing with lips touching the foam pop filters on their microphones. Use direct boxes or guitar simulators instead of mics.
Direct boxes give a clean, tight sound since they pick up no background noise or leakage.
This keeps the muddy-sounding bass notes from bouncing around the room. Sometimes you can pick up two instruments with one mic. Boost the bass and treble a little. Another mic goes in the kick. While listening to the mixer output, bring up the bass-player's fader and ask him or her to play the loudest part of the song.