I will tell you. Case A and case B.
In case A she is in a difficult situation and her faith weakens sometimes. But she knows something is wrong with her, and she still prays and confesses her weaknesses to God. She continues to respect me as the head of the household, asks me spiritual support and advice and she is honest about her feelings. She tries hard not to commit sin and while she may fail sometimes, she knows it is not good and wants to stop. She is weak and human but she wants to be more like God. In this case, not only would I remain with her. I would feel compassion for her, pray deeply for her healing as if she was my treasure in life, support her everytime and be there for her when she needs. She does not deserve the pain of being left alone in any way.
On the other hand, in case B she becomes a different person. She consciously begins to lose her faith and she finds nothing wrong with that. She does no longer pray confess and read the Bible and she loses her respect for me as the head of the household. She does not talk about it even if I find out about her and she will often lie to me. She starts to behave in a sinful way (smoking, drinking *, drugs, looking porn, masturbating or even cheating etc) and doesnt find anything wrong in her behaviors. In this case, I would first warn her about how she is going the bad path and displeasing God and her husband. I would also pray deeply for her healing and reconciliation with God. If after a time she has not been healed and continues to be sinful with no indication of change, I would have to leave her not to risk the future of myself and my children. If she however truly shows understanding and recognizes her problem, I will help and support her in accepting and reconciliating with God, as well as with me. This way, we would head to case A.
* Asterisk: As for the drinking part, this is only valid in case I stopped being drunk myself.
Seems like what distinguishes those two cases is your evaluation of her motivation and if you think she's honest and obedient enough with you. And it's a really murky distinction that makes it sound like (according to your definitions of sin) if she gets a tattoo and sees nothing wrong with it she needs a lecture and to come around to your point of view or else she's an unrepentant non-Chrsitian, but if she keeps cheating on you but knows its wrong and really wants to stop cheating on you then you'll stick around and support her. We can all agree that honesty and trust are essential for a relationship, but feeling like you're going to get a lecture and be told you're wrong everytime you disagree with your spouse tends to cause people to not discuss things with their spouse.