K, I'm gonna throw my two cents in, just for fun. This isn't the exact same situation, but I think it has sort of the same principle. So, at my school there were four fraternities, and four national panhellenic sororities. Then comes along Sigma Alpha - a professional sorority. And there was confusion as to what to do since we had never had a non-IFC, non-NPC GLO on campus.
So... we felt they should affiliate with someone in order to enjoy the rights and privileges that Greeks are afforded and sponsor on campus. We invited them to NPC, which didn't end up working out since they must allow in men, and we cannot, but they ended up affiliating with our President's Roundtable, which includes a representative of every Greek/Greek-related organization on campus. A drawn-up agreement exists between us and them... and now we're all happy.
The purpose of all this was to be inclusionary - not to try and force our system on the ladies of Sigma Alpha. We wanted them to be active, involved Greeks on campus, and it's working.
Why is there such a divide between multicultural/ethnic sororities and the more traditional sorority life? We're more alike than different, I'd say. For schools that have separate councils, is there an organization such as our President's Roundtable that your GLOs get together to at least discuss your respective organizations/situations?