Let's say an Eastern European woman is in Israel on a work visa and a Jewish man, for whatever reason, wants to marry her. Since it is practically speaking illegal to mix in Israel, could they secretly run off to America, and get married? And if so, would that marriage be recognized in Israel, even though they could not get legally married in Israel itself?
First, there is no prohibition against a Jewish man marrying an Eastern European woman in Israel. If they are both Jewish, the marriage can be performed in Israel. If they are not the same religion, then if they get married in Israel, the government will not recognize the marriage. However, Israel recognizes inter-faith marriages performed outside Israel. The marriage does not have to be secret, and it can be performed in Cyprus as well as America.