Is it permissible for a husband to take back his irrevocably divorced wife (divorced on the basis of mutual release/financial settlement - ṭalāq bā’in ‘alā al-ibrā’) during her ‘iddah by saying, "I have taken you back, and you are now my wife, and I call upon Allah to witness that," without an officiant (ma’dhūn)? And is she considered his wife thereby, given her need for a new marriage contract?

If a man divorces his wife with a revocable divorce (طلاق بائن) in exchange for her release (على الإبراء), she is not lawful for him until a new contract (عقد جديد) is made. Therefore, the phrase mentioned does not return the wife to his marital bond, because her divorce is irrevocable, and the contract must be renewed through her guardian (الولي), in the presence of two witnesses, and with a new dowry.