LSD's work "opposite" of the SH-AWD system. The phantom LSD limits how much the tire with less traction slips under power by applying pressure to the spider gears in the tranny when there is a difference in speed detected (mechanically) between the left and right drive wheels. The bigger the difference in speed, the more pressure (friction) it applies to the spider gears. Usually the inside drive wheel is the one with less traction.
The SH-AWD system, on the other hand, applies more torque to the outside rear wheel even when the wheels aren't slipping under acceleration. The rear differential of the SH-AWD system has two clutch packs; one for each rear wheel (there is a third clutch pack that determines the amount of torque applied between the front and rear wheel sets). On right turns, it will let the (rear) right clutch pack slip more to apply more torque to the left rear wheel and on left turns, it lets the (rear) left clutch pack slip a little more to apply more torque to the right rear wheel.
You can think of the LSD as a mechanical device (no electronics) that tries to keep the left and right wheels turning at nearly the same speed and the SH-AWD system taking that one step further by adding electronics and computer sensors.
