Is the رجعة (return to marital union) considered valid after divorcing the wife with a revocable divorce (طلاق رجعي) that becomes an irrevocable lesser divorce (بائن بينونة صغرى), and then taking her back during the iddah (waiting period) without her consent and without engaging in intercourse, while she remains in her family's home? And is it permissible for her to marry another man after her iddah ends?

If this divorce is the first, it is a revocable divorce (ṭalāq rajʿī), and the fact that it occurred at the officiant's (maʾdhūn) or was described as irrevocable (bāʾin) does not negate its revocable nature. The husband has the right to take back his wife who was revocably divorced, even if she does not consent, based on the Almighty's saying: "{And their husbands have more right to take them back in that [period] if they wish for reconciliation.}" So, if the husband says: "I have taken you back into my marital bond," this return is considered valid, and the wife is considered to be within her husband's marital bond even if she is in her family's home. The waiting period (ʿiddah) has no effect after he has taken her back into his marital bond, and she is not permitted to marry another man. It is not obligatory to document the return or to have witnesses for it, but it is better to prevent disputes.