Suppose that Brazil steps up relations with the French starting around the 1840s, because its relations with Great Britain are deteriorating on account of the slavery situation. And suppose that Maximilian (the brother of Hapsburg emperor Franz Josef) still gets married to Princess Charlotte (or Carlota) of Belgium in 1857 as in OTL. Under such circumstances, would the French have aimed for Brazil instead of Mexico in the early 1860s in setting up a sphere of influence in Latin America? Would Maximilian have accepted offers in Brazil (at first becoming Heir Apparent) and rejected offers in Mexico?
If so, then I see quite a few effects. To start with, in Brazil, Maximilian becomes Emperor of Brazil before long, and the Empire of Brazil is much prolonged (so no republic as early as 1889). Max survives beyond 1867, when OTL he is killed in Mexico after his defeat. Brazil might even today be a constitutional monarchy, and a little more prosperous than OTL. In Mexico, Benito Juarez would have remained as the undisputed president through the 1860s all the way to his death in 1872; there would also be no Cinco de Mayo holiday. The French military presence in Mexico (along with those of the British and Spanish) is confined to debt-collection exercises at the port of Veracruz. Are these things fair to say? Any more butterflies?