seems like you have 2 options....

accept the fact that "real" monogamy is quite possibly never going to happen and continue the relationship

end the relationship over the lies and cheating and walk away.

I can only tell you what I would do. it involves a good pair of nike shoes. life is too short to deal with someone who doesn't value you enough in a relationship to not cheat. period. the lying just makes me move quicker. this is not something that will end right now...there is a pattern.