She does not sound poly to me because poly is based on open communication with full knowledge and consent of all parties. She is not even on the swinger side of the spectrum since that involves trust as well.
It seems she gets her thrills from cheating itself. Agreeing to be non-monogamous may only be treating the symptoms. I would be more worried that she is self-destructive and does things on purpose to hurt the relationship.
I don't think allowing her to have sex in front of you will cure her desires. I think tha is you both gave each other the freedom to date, you may have a better chance. However, I don't know how you fix the honesty and trust.