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90 wt gear oil from any parts store and a bottle of limited slip additive from the GM parts counter required. Buy 2 bottles in case you need more than 1.
I'm running Schaeffer's after folks with noisy GM truck diffs had success in quieting them down. It helped quiet my diff. Their website has a list of sources.
My '02 Z rear end has been rebuilt twice under warranty and the dealer uses synthetic with no additive. They say the additive is not needed with synthetic lube. Lots of other noises but never had any posi clunking.
GM gear lube and GM limited slip differential additive from my local Chevy dealer. It is easier if you have a part # for both products. Some of these parts depts have no clue.
Castrol gear lube from AutoZone. Right on the same shelf they also have the Posi additive for a few bucks and it states right on the tube that it's compatible with the GM # (which I can't think of off the top of my head). Nice just to get it all in one place.