Is it permissible for a married man to marry another woman who embraced Islam on the condition that he marry her, without the knowledge of his first wife, and without documenting the marriage?

It is not permissible for young men and women to get to know each other, even if the intention is marriage, due to the strife and corruption that may result from it. It is permissible for you to marry this woman if she converts to Islam or if she is a chaste woman of the People of the Book, and if you are able to bear the costs of marriage and deal justly between your two wives. It is recommended to marry her if you hope for her conversion to Islam. Your first wife's knowledge is not a condition for your marriage to her, nor is documentation of the marriage required, though it is desirable for safeguarding rights. For you to marry her while she is a woman of the People of the Book, it is a condition that it be through her guardian. However, if the marriage is after her conversion to Islam, the Muslim judge will conduct the marriage contract if she does not have a close Muslim guardian. You should strive to invite her to Islam through some Muslim women, as there is a great reward in that.