What is the ruling on a couple remaining together if the husband falsely swore to his wife by Almighty Allah that he was not in a relationship with another woman, and he repeated the oath by saying: "If you find that to be true, then she is forbidden to me until the Day of Judgment," placing his hand on the Qur'an, and then he confessed to his lie?

The original ruling is that it is permissible to lie to one's wife if it strengthens the marital relationship, as stated in the Hadith of Umm Kulthum. However, if the relationship he swore to abstain from was unlawful, then he is sinful because of it. As for his statement that his wife is permanently forbidden to him, this is a conditional dhahir (a pre-Islamic Arabian custom akin to divorce). If the condition upon which he made the dhahir is met, he is obligated to perform the kaffarah (expiation) for dhahir. His statement "until the Day of Judgment" has no effect; rather, his wife becomes forbidden to him, similar to a woman upon whom dhahir has been pronounced, until he performs the expiation.