I was thinking of the following "what if" scenarios:
US - French War circa 1860's?
The other two sounded very plausible, but this one is very (very!!!) unlikely - primarily because the US Army was busy styling itself exclusively on that of France at the time! I think even the relevant US drill books were translations of their French equivalents!
But then, America did already have "history" of biting the hand that fed it......

British intervention in the franco prussian war 1870 ....
Although the 1860s had seen an anti-French invasion scare, leading to the construction of several forts along the south coats of England, it had also seen Prussia invade and annexe Hannover, so neither side - or both! - was particularly appealing to interventionists. However, there were several unofficial British-raised ancillary units on both sides, particularly ambulance columns; in one, serving with the Prussian army, was a British Army surgeon named William Manley, who became the only person to be awarded the Victoria Cross (Maori War) and the Iron Cross (FPW).