I can certainly empathise with your resistance to label yourself. I don't find it adds any value in my life to stick a tag on, except when I really am "confused" and the label helps me find people going through the same struggles. But I find that in those cases, the label is more of a "signpost" that I use as a stepping stone towards better understanding, rather than something that becomes part of my identity.
For example, I recently heard the term "gray-A" and asked what it meant. I was told it referred to people who are more-or-less asexual -- that when it comes to sex, they can take it or leave it -- but they're not wholly adverse to sex and they can enjoy it without needing it. That description really resonated with me, and the term gave me something to Google so I could learn more about it. But after going though that process, I didn't feel the need to identify as gray-A, it was just enough to know that it was "a thing" that I could relate to.
I see nothing wrong with just saying that you don't resonate with any labels, but that you'd be happy to explain your relationship dynamics to someone who's interested (if you are interested, of course; otherwise, choose your version of MYOB). If you just need a quick reference to get the ball rolling, something like "poly-ish" could be useful.
As an aside, nothing prohibits people who identify as mono from coupling with people who identify as poly. We are not defined by the people we date. But whatever reasons you have for not identifying as mono are perfectly valid.
Gralson: my husband (works out of town).
Auto: my girlfriend (lives with her husband Zoffee).
The most dangerous phrase in the English language is "we've always done it this way."