Keep in mind that the slipprey, low friction, gear oils that promise "more HP to the rear wheels" also diminish the effectiveness of the posi clutches, as do the "anti chatter" posi additives.
Lo friction of the gears and bearings, is also lo friction on the posi plates. Exactly opposite of what the posi needs.
Rather than adding additives, let's look at why the posi chatters going around corners at low speeds. The clutches only get lubed at very low speeds and while turning corners because the ring gear "pumps" the fluid out and away from the posi clutches while driving above about 20 mph.
Posi chattering? Simple solution. Drive for 20 miles to warm up the gear lube. Then do two left complete 360° turns, and two complete right 360°turns at a slow walking speed. This pulls warm lube in-between the clutches as they rub together during the turn because the fluid is not flung away from them as when driving. Most chatter is noticed on cars that are hiway driven. The the 2 turns each way cures that without diminishing the posi action.
Don't do this when driving to or at the dragstrip! We want as much friction on the posi clutches as possible there!
The pressure angle of the spider gears in a posi are much "flatter" to promote higher side loading to the clutches on acceleration. That's why when stuck, applying the parking brake (with a posi) lightly, helps the torque bias to the spinning wheel.
Detroit lockers and worm gear posis do not have clutches, so synthetic fluids are fine with them.
TIMINATOR