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Why would you want to move an axis system from a part? You would always move the part from an axis system. To make you understand it is a bit like saying "i want to move the location of the wing on an aircraft, but rather than move the wing with reference to the aircraft, I will move the aircraft with reference to the wing"
If you are not fussed about consistency, I would do as MrCATIA said and simply paste the product in a new product and put in a new part at the origin of the new product with a defined axis system. Than I would move my old product with reference to the axis system in my new part. This is assuming your X,Y & Z is not the same as the compass.
1. Insert New Part into Product
2. Prompted Origin answer NO
3. Create Fix constrain on this new part. (Delete any Fix constrain in your product)
4. Mark all Parts inside the Product without NewPart and make FixTogether constrain.
5. Make distance constrain with NewPart as you wont.
6. Update Constrains
7. Delete NewPart (or not)
This should move all product to new position.
Because global origin is invisible, inserting new part his plans are global origin, so you can see where it is.
Fixing together all parts they will move all together, but you cannot move manually any of parts. Update FixTogether will not move this part back.