Is it considered permissible to have sexual relations with a married Christian woman, with the knowledge of her family, her husband, and some Filipinos, and given that there is no divorce in her country's law?

There is no might or power except with Allah. The questioner must repent immediately and leave this woman, especially since she is still married, and having intercourse with her is even more forbidden.

Her mere separation from her husband does not mean she is divorced, and it is not permissible for anyone to marry her until the divorce is confirmed.

If the woman is not married, then marriage can only take place under Sharia conditions, such as the guardian's permission and the presence of witnesses. The mere knowledge of her family does not make her permissible for him.

In addition, it is a condition for a Muslim man to marry a woman from the People of the Book that she be chaste, and it is not permissible to marry an adulteress.