What is the religious ruling on a woman remaining in an illicit relationship and feigning marriage without a contract, what is the duty of her two daughters towards her, and do they incur sin because of this relationship?

The way to deal with a mother who commits adultery is by advising her and preventing her from it, while informing family members who have influence over her to help her desist from forbidden acts. The mother's right remains, so she must be treated with kindness, dutifulness, and excellence, and be prevented from evil. A marriage contract does not require notarization in court; the guardian's consent, offer, the husband's acceptance, and witnessing are sufficient. The daughter's husband should not be informed of the mother's bad behavior unless it will help resolve the issue or influence her.