Religious institutions.
Or suppose we are talking about an infertile couple.
Just to isolate the question toward the lives of the parents themselves. Is this the way of the Torah?

I don’t know how to answer “the way of the Torah.” There is no prohibition, but usually strong fear of Heaven will be hard to produce from such a home. There are also questions of trustworthiness and kashrut and the like.

It should be forbidden because of: “Neither shall you bring an abomination into your house”
🙂

Regarding a couple in which one of them changed status (became religious / went off religion), would you recommend separating for the above reasons?

In a case where we are talking about a woman who fits the definitions required for public Sabbath desecration (or who is simply an apostate with respect to the entire Torah), would there not be a prohibition because of the distancing rules that are the same as the distancing rules from gentiles?

That is an interesting point. I think that if we are dealing with a wicked woman who knows her Master and rebels against Him—then yes. But in the simple sense, this is not wickedness but a different belief. And someone who holds a different belief is not wicked, and the laws of distancing do not apply to her. And if we are talking about his wife, it seems there is no room at all for laws of distancing, though when he marries her and knows that this is her nature, he is placing himself into a situation of duress, and this requires further study.
As an aside, I will mention something I once wondered about: is a Sabbath desecrator forbidden to drink his own wine? Meaning, if he drinks wine that he himself poured or opened, he would then also be culpable for that. And logically it seems obvious to me that he is not.