This is what I mean by fake-sovereignty

Under the Balfour Declaration of 1926, dominions were proclaimed to be "equal in status, in no way subordinate one to another in any aspect of their domestic or external affairs, though united by a common allegiance to the Crown"[31] and the monarch is the leader "equally, officially, and explicitly of separate, autonomous realms".[33] Andrew Michie wrote in 1952 that "Elizabeth II embodies in her own person many monarchies: she is Queen of Great Britain, but she is equally Queen of Canada, Australia, New Zealand, Pakistan, South Africa, and Ceylon ... It is now possible for Elizabeth II to be, in practice as well as theory, equally Queen in all her realms."