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If it was me, the things I'd suspect in order are:
Parking brake
limited slip friction modifier deficiency
springs/suspension
If you have already checked the first two....then you might try mounting a video camera under the bumper, aim it at the location where the axle is connected to the leaf springs and then record a back up sequence. If there is any movement then you might have a broken locating pin on the spring pack.
I would think if the rear diff is low on friction modifier
it would chatter making turns on a hard surface.
Maybe, or what about, if the rear-ends' been re-geared but not shimmed correctly. Would it last this long or be heard going down the road? He can hear it backwards since speed isnt in equation?
Mine is doing the same thing. I would be willing to bet that it is just the rear end needing the friction modifier as stated. It seems to do it worse in reverse compared to moving forward. The last time mine did it, the modifier was added and it went away. I am getting ready to replace the rearend oil so that is why I haven't added the modifier to mine yet.
Buy the 4oz bottle and add a few ounces to start. Then if it is still doing it, add another once at a time untilled it goes away. You can do a search on the subject and you will find all the details.
The additive seemed to help.
Have driven 200 miles since I added it and its not half as bad.
Can still feel it however , but at least I know what it is now.
Mine was doing the same thing when I first bought it. I changed the rear end fluid and added the friction modifier. It took a while (maybe a month or two) before it was back to normal.