A semi-plausible one (but not in the 1880s-1900s) would be that a much more successful Pacific War for Japan means that the USA comes out of the war worse for wear than their original got-off-scot-free state. The UK would also have to come out of it worse than in OTL. They basically form a personal union, with two systems coinciding with one-another, in order to survive the immediate results of a shared blow to their economies and infrastructure.
The USA and British Empire are merged and the new entity is renamed the United Commonwealths, or the United Anglo-American Federation.
The UK is a Commonwealth that manages its state and has a Prime Minister and Monarch, independent and the leader having equal representation alongside the co-leader, the American President. These two will be the heads of a cabinet that works semi-separately from the British and American Cabinets, but makes decisions affecting the whole UC/UAAF.
The British Monarch still has their symbolic rule in Britain and the Crown Realms, with her zone of authority residing in the Imperial Commonwealth. Americans that come over to reside and gain internal citizenship of the Imperial Commonwealth are now able to be given noble titles if they so wish. To keep the Americans happy, this does not extend outside the Imperial Commonwealth. The American Commonwealth is the twin entity of the Imperial Commonwealth. They vote for their President, and keep their republicanism.
The British Parliamentary system stays in the Imperial Commonwealth with the American Congress not affecting their decision, but the Parliament can vote for the British members of the Higher Council (the one headed by the PM and POTUS). This is the same for the Congress, who elect the American representatives of Higher Council.
The places that are left of the Imperial Commonwealth (ones who did not become independent, like India and other places) get equal and fair representation in the Higher Council and the rest of the UC/UAAF.