I have a corgi/basenji mix if you want to look at his album on my profile :) He isn't tricolor though, he is the standard orange and white.
He definitely exhibits a lot of the basenji traits: He's VERY clean (if he gets his feet wet or dirty he will spend an extended amount of time cleaning them until his paws are sparkling white again), independent, "cat like," he likes to watch TV, he never smells even when wet, and although he can bark he makes lots of odd basenji like yodeling noises. On the other hand he certainly got the corgi herding mentality.
I've found this mix to be ideal. I grew up around purebred corgis that were around horses all the time and although I loved them they weren't as relaxed as our mix and defintely not as clean!