Is it permissible for a husband to take back his divorced wife with a formal agreement during her waiting period ('iddah) without a new marriage contract?

If a man divorces his wife with a revocable divorce (ṭalāq rajʿī), he has the right to take her back (rajʿah) before the expiration of her waiting period (ʿiddah) without the need for a new contract, or her consent, or the consent of her family. However, since the divorce is officially registered, it is necessary to officially register the return (rajʿah) to safeguard the wife's rights.