All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It is not a condition for the divorce to be effective to have witnesses for it; rather, divorce takes place once the husband utters it. Having witnesses for divorce is not obligatory, let alone being a condition for it to be effective. Rather, it is only desirable to have witnesses for divorce according to the view of the majority of the scholars as we already clarified in Fatwa 133297.
Therefore, your wife has become irrevocably divorced from you and she is not permissible for you unless she marries another man a serious marriage and he consummates the marriage with her and then he divorces her or dies. For more benefit on the kinds of divorce, please refer to Fatwa 82541.
Indeed, having sexual intercourse with her after the third divorce had taken place is a major sin, so you are both obligated to repent. Your wife is also obligated to repent for having committed Zina (fornication or adultery) with that man. For more benefit, please refer to Fatwa 86527.
Moreover, she should protect herself by observing Hijab and avoiding staying in seclusion with non-Mahram men or mixing with them or other matters that could lead her to commit Zina again.
If her waiting period expires, it is permissible for her to look for a pious man who would marry her; however, this marriage should not be in order to make her lawful for you. This is because both the husband who seeks that his wife be permissible for him again by her marrying another man, and the man who marries her for this purpose are cursed. ‘Ali narrated that the Prophet said: “Curse be upon the man who marries a divorced woman with the intention of making her lawful for her ex-husband, and the one for whom she is made lawful (i.e. ex-husband).” [Abu Daawood, At-Tirmithi and Ibn Maajah]. For more benefit, please refer to Fatwa 92332.
Finally, it should be noted that a person who does not know (the ruling on something) must ask the scholars before acting especially in regard to the issues of marriage so that a person would not commit what is forbidden while not realizing it. Allaah Says (what means): {…ask the people of the message [i.e. former scriptures] if you do not know.}[Quran 16:43]
Allaah Knows best.