The washer drain goes into a piece of PVC pipe that has been routed
thru the partition wall, behind the toilet, and into a 6" or so
vertical vent pipe in the corner.

It's connected to the vent pipe (T or Y and a reducer)? Or someone just
hacked a hole into the vent and shoved the PVC into it? Either way why
would anyone expect that sewer gas *woudn't* come out? Put a trap in the
line.