Is it permissible for a woman to marry and register the marriage at the real estate registry instead of the court to prevent her ex-husband from knowing about the marriage and to protect her children? And what is the ruling of Islamic law on that?

A woman is not permitted to conceal her marriage for the purpose of maintaining custody of her children, because Islamic law invalidates a woman's custody if she marries a man who is a stranger to the child, based on the Prophet's (peace be upon him) saying: "You have more right to him as long as you do not marry." It is not permissible to resort to stratagem to circumvent this ruling. The marriage is valid if its conditions are met. In the event of a dispute over custody, the Sharia (definition) court shall decide the matter.