What is the ruling on having children with a foreign woman and cohabiting with her for a long period without a marriage contract? What is the ruling on the relationship of these children with his children from the second wife? And what is the ruling on that in terms of inheritance?

If a man has sexual relations with a woman without a marriage contract and she gives birth to children, this is zina (fornication/adultery), and the children are not attributed to him, nor do they have a right to his inheritance. However, if he had contracted marriage with her, even if the contract was not registered in the courts, the children are attributed to him, and they are paternal half-siblings to his children from the second wife, regardless of whether the contract was valid or invalid, as long as he believed it to be valid. Ibn Taymiyyah said: "Muslims are in agreement that every marriage which the husband believed to be permissible, if he engaged in sexual intercourse through it, then his child born from it is attributed to him, and they inherit from each other by the consensus of Muslims, even if that marriage was invalid in reality, by the consensus of Muslims."