Is it permissible for a man to reconcile with his wife whom he has divorced, and with whom he has consummated the marriage, without the knowledge of her family, given that the divorce certificate states that he has not consummated the marriage with her?

A husband may take back his divorced wife with whom he has consummated the marriage, or with whom he has had a valid private seclusion (khalwa sahiha), as long as she is still within her waiting period ('iddah), and this applies to the first and second divorces, without needing a new contract or dowry. As for the wife with whom he has not consummated the marriage and has not had a valid private seclusion, she has no waiting period, and he may take her back with a new contract and dowry. The divorce certificate does not change the Shari'ah ruling.