What must a husband do after telling his wife, "You are divorced if you don't say such-and-such, or if you don't do such-and-such, I will not sleep with you," and then the wife does what was required of her?

What was uttered by the husband is an oath of divorce. If he intended it to take effect immediately and the wife delayed in fulfilling it, then he has broken his oath and divorce is binding upon him according to the majority of scholars. He may take her back if it was the first or second divorce. Sheikh al-Islam Ibn Taymiyyah is of the view that an oath of divorce is an oath for which expiation (kaffarah) is due if it is broken.

However, if the husband intended a delay in fulfilling what he swore upon, or if he did not intend divorce by the كناية (euphemism/allusion), then nothing takes effect. It is obligatory to be cautious about using words of divorce, whether in seriousness or in jest.