What is the ruling on writing the marriage contract for a woman who is customarily married, and whose husband has abandoned her while she is pregnant, in order to protect her reputation? Is it permissible to consummate the marriage with her after childbirth to protect her during delivery, or is it obligatory to file a case of adultery due to the lack of proof of paternity? What is the situation in this case?

This marriage appears to be a marriage that fulfills the conditions, but it is not officially documented. If that is the case, then this woman is married, and it is not permissible for anyone to contract marriage with her due to the existence of a legitimate impediment (her being already married). If this marriage were to occur, it would be void. The woman can prove it with witnesses or by those who heard of the marriage, in order to establish the lineage of her child.