Are a couple who did not document their marriage in the country of residence and were judicially divorced in a Western country after the husband's attempts at reconciliation and threats of divorce, then reconciled and returned to live together without renewing the contract, considered to be committing adultery Islamically?

If you did not issue an oral or written divorce, and you did not authorize the lawyer to divorce, then divorce does not occur, even if you authorized him to threaten with divorce, because a threat is a warning by which nothing takes effect. The wife's distance or desire for divorce does not harm, as the marriage remains valid. It is advised to treat the wife kindly and resolve disputes, as Allah Almighty says: (And live with them honorably), and His saying: (And due to them is similar to what is expected of them, according to what is reasonable. But the men have a degree over them), and the saying of the Prophet, peace be upon him: (Treat women kindly), and (The best of you are those who are best to their wives, and I am the best of you to my wives), and (If I were to command anyone to prostrate to anyone, I would have commanded women to prostrate to their husbands, because of the rights Allah has given them over them), and (If a woman prays her five [daily prayers], fasts her month [of Ramadan (definition)], guards her chastity, and obeys her husband, she will enter Paradise from whichever of its gates she wishes).