I turned my simple 90° indexer into a light slip clutch and to me it seemed very suitable for application in a limited slip differential. I replaced the angle element with square mid-section with a 2L axle connector so it has 8 preferred positions (45° steps) instead of the 4 in the 90° indexer.
I played a little with it and even in a small light-weight push-along car you can sense the straight-line-stability and the clutch actually starts slipping when pushing the car through turns. The white silicon band can be replaced with a red one wrapped around the limiters (2L liftarms) twice to make the clutch tighter. I think this approach gives a very realistic effect, or at least an effect that is realisticly proportional to Lego vehicles, also motorized vehicles. But you will have to find out that for yourself.
EDIT: As has been pointed out below this does not really qualify as LSD. The friction clutch is always engaged and for it to be a LSD the clutch should only be engaged when one of the output axles slips.
LXF-file here.