If a Muslim man has divorced his wife, but then has a change of mind, he cannot simply remarry her even if both want to start over again. Islamic law stipulates that the wife must first marry and have intercourse with another man, and who has to divorce her again, before it becomes lawful for the original couple to remarry.
The man who marries a divorced wife in order to make her lawful for her former husband again is called muhallil, which literally means, "One who makes lawful."
The basis for this law is found in the Qur'an:
Divorce is twice; then honourable retention or setting free kindly. It is not lawful for you to take of what you have given them unless the couple fear they may not maintain God's bounds; if you fear they may not maintain God's bounds, it is no fault in them for her to redeem herself. Those are God's bounds; do not transgress them. Whosoever transgresses the bounds of God -- those are the evildoers. If he divorces her finally, she shall not be lawful to him after that, until she marries another husband. If he divorces her, then it is no fault in them to return to each other, if they suppose that they will maintain God's bounds. Those are God's bounds; He makes them clear unto a people that have knowledge. S. 2:229-230
Muslim tradition specifies that this cannot be a pro forma marriage (i.e. they cannot only marry on paper, not only register the marriage), but the woman must engage in sexual intercourse with the man before she can return to her former husband:
Yahya related to me from Malik from al-Miswar ibn Rifaa al-Quradhi from az-Zubayr ibn Abd ar-Rahman ibn az-Zubayr that Rifaa ibn Simwal divorced his wife, Tamima bint Wahb, in the time of the Messenger of Allah, may Allah bless him and grant him peace, three times. Then she married Abd ar-Rahman ibn az-Zubayr and he turned from her and could not consummate the marriage and so he parted from her. Rifaa wanted to marry her again and it was mentioned to the Messenger of Allah, may Allah bless him and grant him peace, and he forbade him to marry her. He said, "She is not halal for you until she has tasted the sweetness of intercourse." (Malik's Muwatta, Book 28, Number 28.7.17)
Yahya related to me from Malik from Yahya ibn Said from al-Qasim ibn Muhammad that A'isha, the wife of the Prophet, may Allah bless him and grant him peace, said when asked whether it was permissible for a man to marry again a wife he had divorced irrevocably if she had married another man who divorced her before consummating the marriage, "Not until she has tasted the sweetness of intercourse." (Malik's Muwatta, Book 28, Number 28.7.18)
The man who makes the woman lawful for her former husband is called Muhallil:
Yahya related to me from Malik that he had heard that when asked whether it was permissible for a man to return to his wife if he had divorced her irrevocably and then another man had married her after him and died before consummating the marriage, al-Qasim ibn Muhammad said, "It is not halal for the first husband to return to her." Malik said, about the muhallil, that he could not remain in the marriage until he undertook a new marriage. If he had intercourse with her in that marriage, she had her dowry. (Malik's Muwatta, Book 28, Number 28.7.19)
This Islamic law stands in stark contradiction to the Biblical command:
"If a man marries a woman who becomes displeasing to him because he finds something indecent about her, and he writes her a certificate of divorce, gives it to her and sends her from his house, and if after she leaves his house she becomes the wife of another man, and her second husband dislikes her and writes her a certificate of divorce, gives it to her and sends her from his house, or if he dies, then her first husband, who divorced her, is not allowed to marry her again after she has been defiled. That would be detestable in the eyes of the LORD. Do not bring sin upon the land the LORD your God is giving you as an inheritance." Deuteronomy 24:1-4
"God says, 'If a husband divorces his wife and she goes from him and belongs to another man, will he still return to her? Will not that land be completely polluted? But you are a harlot with many lovers; Yet you turn to Me,' declares the LORD. Lift up your eyes to the bare heights and see; Where have you not been violated? By the roads you have sat for them Like an Arab in the desert, And you have polluted a land with your harlotry and with your wickedness." Jeremiah 3:1-2
What God calls detestable, the Qur'an commands Muslims to perform. It is one of many reasons the Qur'an does not originate from the same source as the Bible. It is not from the God of the Bible. See also: Does Allah Command Evil?, A strange divorce and remarriage law in the Quran
Even from the time of Muhammad, we have reports about some of the bad consequences resulting from this Islamic regulation:
Narrated 'Ikrima:
Rifa'a divorced his wife whereupon 'AbdurRahman bin Az-Zubair Al-Qurazi married her. 'Aisha said that the lady (came), wearing a green veil (and complained to her (Aisha) of her husband and showed her a green spot on her skin caused by beating). It was the habit of ladies to support each other, so when Allah's Apostle came, 'Aisha said, "I have not seen any woman suffering as much as the believing women. Look! Her skin is greener than her clothes!" When 'AbdurRahman heard that his wife had gone to the Prophet, he came with his two sons from another wife. She said, "By Allah! I have done no wrong to him but he is impotent and is as useless to me as this," holding and showing the fringe of her garment, 'Abdur-Rahman said, "By Allah, O Allah's Apostle! She has told a lie! I am very strong and can satisfy her but she is disobedient and wants to go back to Rifa'a." Allah's Apostle said, to her, "If that is your intention, then know that it is unlawful for you to remarry Rifa'a unless Abdur-Rahman has had sexual intercourse with you." Then the Prophet saw two boys with 'Abdur-Rahman and asked (him), "Are these your sons?" On that 'AbdurRahman said, "Yes." The Prophet said, "You claim what you claim (i.e. that he is impotent)? But by Allah, these boys resemble him as a crow resembles a crow." (Sahih Bukhari, Volume 7, Book 72, Number 715)
Whatever the reasons for their marital disagreements, the man was beating his wife (which the Qur'an allows, and which seems to be of no concern to Muhammad), but based on this law, Muhammad demands from her to have intercourse with this man, even though this may not be possible. Men can become impotent for many reasons. Perhaps 'AbdurRahman was able to perform intercourse in earlier years, but he had become impotent later. Having two sons does not prove anything about a man's present ability for intercourse. Muhammad was apparently unaware of this basic medical fact. Whether she was lying, or he was lying, it is obvious that this was not a happy marriage. Muhammad demanded that two people who had no love for each other, have intercourse before the woman could remarry her former husband, assuming that he wanted her back.