What is the ruling regarding a wife leaving the marital home, her children, and her family, and marrying another man on paper? And what should the first husband do if she returns in such a case?

If a woman marries while still married to another man, her marriage is invalid, and the second husband must separate from her immediately.

Then, upon her return to her first husband, he must refrain from intercourse with her until the istibra' period concludes. This period involves her menstruating three times, or three months if she does not menstruate, or delivering her child if she is pregnant. This waiting period is to ascertain the clearness of the womb from pregnancy. The wife's claim that she has not been intercoursed with is not to be accepted.

Some scholars are of the view that istibra' is achieved with one menstruation, as this suffices for the istibra' of slave women and umm walad (a slave woman who has borne a child to her master).

If the wife has committed what is mentioned in the question, it is recommended to divorce her, and it might even be obligatory, especially if she has neglected her obligatory duties towards Allah or if she is not chaste.

Finally, we emphasize that a woman leaving her husband's home without his permission and without a legitimate Islamic excuse is a sin and constitutes nushuz (disobedience), which forfeits her right to maintenance.