For a different take, it should be possible to set up a "little brother" type of dynamic, where the dominant power shares close cultural ties without actually holding any kind of real power - there'd be some level of domination just due to the size different, but nothing overt.
*USA and *Haiti, with a PoD of the early leader in the Haitian Revolution deciding to make hostages out of the Europeans, rather than killing most of them? Avoiding the massacre prevents *Haiti from being a pariah state, may plausibly result in an early recognized independence (convince Britain and the US to recognize the nation, and they can prevent Napoleon from landing troops), could butterfly into gradual manumission in the border states (Virginia still came close until Nat Turner; without the example of a terrifying slave rebellion, that push is much stronger), and yet there's no reason that the US should really try to annex the nation.
Cultural dominance is guaranteed, of course, but with the free/slave state balance needing to be maintained until after the Civil War there's not going to be a strong push to make *Haiti a state, and if the revolution doesn't completely destroy the economy, Britain as a secondary trading partner helps avoid a banana republic situation. You could well get *Haiti in economic orbit around the *USA without political control.