I just "googled" the term "biamory" and found that it is in increasing use.
Here's an interesting discussion on the matter (Ther are many):
"It seems to me that the need to define bisexuality (or sexuality in general) by willingness to enter into long-term relationships is probably a holdover of the attitude that one should only have sex inside of marriage. Sex outside marriage (or at least a long-term relationship) "doesn't count." Of course, what it boils down to is that we're discussing two separate subjects entirely (sex, and long term relationships) and getting frustarted at out inabilities to ram both into the same pigeonhole ("bisexuality"). There is a term, "biamory," which specifically describes the willingness to be in a loving relationship with either gender."