Is it permissible to consort with European women without marriage, given that they are polytheists, and in consideration of the conditions imposed by Muslim families for engagement?

It is Islamically forbidden for a Muslim man to be in a romantic relationship with a woman who is unlawful to him, whether she is Muslim or a disbeliever. This is based on the verse: "marrying [them] in purity, not [engaging in] unlawful sexual intercourse or taking [them as] secret lovers." If he cohabits with her as a husband, this would be adultery. The fact that the woman is a polytheist does not make this permissible for him. Rather, he is obligated to keep himself chaste until Allah facilitates marriage for him, in accordance with the verse: "But let those who find not [the means for] marriage abstain [from sexual relations] until Allah enriches them from His bounty." He must seek help from Allah and exert effort to find a righteous wife.