I'm on the other end of that: I'm the side piece for a married man with his wife's knowledge & permission. Our relationship is completely outside the "norm" so we get to define it however we like and set whatever terms work for all of us. These terms are endlessly negotiable as circumstances change.
Publicly, they are a monogamous married couple. This is the trouble with non-traditional/open/poly/monogam-ish: no one hears about the ones that work because most cannot be open about it for fear of consequences. We only ever hear about the ones that go down in flames and everyone points at the non-traditional aspect as the cause, which may or may not be the case. Compare this to the LGBTQ issue: once upon a time, no one knew anyone who was gay/lesbian... until they realized they already did because coming out became safe(r). Today everyone knows someone who's trans: Caitlyn Jenner. Sooner or later, everyone will realize they already know someone who is not in a traditional marriage and that will no longer be a weird or unusual thing.
You know yourself, you know your spouse and only the two of you can decide if you're willing to take that step and try something really REALLY different. Open/poly/mongam-ish requires vast amounts of communication, deep self-examination, brutally open honesty, and willingness to do the work. When those things come together with people who value each other, it is amazing. Everyone's quality of life increases and there's more support in times of struggle.