Maybe, if we're being idealistic, it needn't be particularly powerful. We could just say that every authority would get their own permutation of the ID, a hash

Maybe, if we're being idealistic, it needn't be particularly powerful. We could just say that every authority would get their own permutation of the ID, a hash of the real ID and the authority's ID or some such, just to make it fairly difficult for two companies to see if their respective IDs are identical. Not perfect, but simple and makes it harder to exploit.