To go back to the original Q... the AWD system transfers power between the front and rear differentials, trying to keep both front and rear wheels spinning at the same speed. Without an LSD, though, all the front or rear power could be directed to the spinning wheels.
If I understand everything correctly (and please correct me if I'm wrong), if your right wheels are on ice and your left are on pavement, without an LSD your "AWD" car would be a "RWD" car. Not Rear Wheel Drive... Right Wheel Drive. All the power would be sent to the 2 spinning wheels and you'd go nowhere fast.
The LSD on the rear will at least turn it into a 3WD car... rear two and the front right, which is better, but not "AWD" imho...
And what's that magic blue stuff that mysteriously goes to the "wheels that grip" in the Subaru commercials? Can I get some of that?