I think the original question posed was. "Do I need to anneal .375 H&H cases?"
The 375 H&H (and it's related cartridge family) headspace on the BELT. So the stretching/annealing question is dependent on the chamber dimensions and the actual cartridge dimensions. Rifle makers want their rifles to be able to digest any cartridge up to max CIP specifications. They ream chambers close to, or at, maximum spec dimensions. Cartridge manufacturers want they cartridges to load into any chamber reamed to CIP specifications, so they tend to make cartridges that will fit into a minimum chamber. Since the headspace specification deals with the belt, meeting that spec woud seem to deal with head spacing issues. But the rest of the chamber - the part where the cartridge might expend on firing could, in fact, be some thousandths of an inch larger than the cartridge. When fired the whole cartridge will expand to fit the chamber. This stretches the cartridge in length and diameter. This stretching work hardens the cartridge and typically causes a thinning of the case wall ahead of the belt, which is where the 'extra' brass is coming from. This is manifested in case head seperations which you do not want.
The designers of the 375 H&H never expected that their customers would be re-using cartridges once the they were fired, so they probably didn't concern themselves with brass life after one use.
Longitudinal stretching can be controlled by resizing the cartridge to headspace on the shoulder of the chamber, like normal rimless rounds. This will help minimize the longitudinal stretching and thinning of the case just ahead of the belt by limiting the distance that the brass will stretch and the resulting displacement of case metal.
Pressure from firing not only stretches the brass, but also causes it to expand outward to the limits of the chamber diameter. Resizing the brass will partly remedy this, but the resizing die cannot resize the small area ahead of the belt because the die has a slight inner taper that cannot reach down onto the face of the belt. BUT, there is a tool made by Innnovative Technologies that can move this critical area back to proper diameter.
Long way around to saying that the life of the .375 H&H brass can be extended significantly but it requires some more tools, care and screwing around to get it done. But with the price of .375 H&H brass it may be worth it to you. If you choose to try to extend the life of your brass it makes sense to anneal the necks as a safeaguard against problems with split necks, etc.