You've heard of a Rum and Coke?
It's commonly called a Cuba Libre, though some argue that this also requires lime juice.
The other day, I picked up a pizza when I picked up my son from work at Mod Pizza, and decided to get some Birch Beer, which is somewhat hard to find, but always stocked there.
When I got home, I poured the birch beer in a glass, and mixed it with some (a lot, actually) dark rum. (No citrus added)
The wintergreen overtones (The oil in wintergreen and birch are almost identical) works very well with dark rum.
My only question now is what to call it.
"Rum and Birch" sounds like some sort of S&M thing.