^^If the Jewish person is Orthodox and wants to be married by an Orthodox Rabbi, then the gentile has to be converted by an Orthodox Rabbi. A Conservative or Reform Rabbi won't do, just as it won't do if you want a free "Right of Return" to Israel, given the stranglehold ultra-Orthodox political parties have on the Knesset.
Of course, these are American permutations (Conservative Judaism and Reform Judaism) of the Jewish faith (maybe Anglo world as well, like Australia perhaps?). They don't apply, to my knowledge, to Europe, nor, I would think, to Israel. So far as I know, for Sephardim, Italkim, Iraqi and Iranian Jews, there's one kind of Jewish. You follow it or you get out of it. I'm speaking in terms of religion, not culture.
There are some fringe minority groups too.
King John probably knows more about it.
I do know that I had a very good friend in university who married a gentile who didn't undergo an Orthodox conversion, and the family sat shiva for him. In other words, they held a mourning ceremony in their home, mirrors covered, clothes ripped and everything, because they considered him dead to them. It was pretty horrifying.
One has to consider too that in the Classical Era, after sometime early in the 300s C.E. intermarriage between Jews and Gentiles was forbidden by the state, and the penalties were extreme. You couldn't get around it by conversion because Jews were forbidden, on pain of death, from proselytizing. They were also prohibited from converting their Gentile slaves. European states enforced those edicts once Rome fell. So the only way it could be done was if the Jewish person converted to Christianity and then married a Christian. The gene flow was thus much more often from the Jewish community into the Gentile one, rather than the other way around. Of course, rapes are another matter, and there is indeed some small amount of East European ydna and non-eastern R1b in the Ashkenazim. Fueled by vodka as many of the pogroms were, maybe the men were incapable, or as has happened in other places and other times, women desperate enough to want to rid themselves of a product of rape, will find a way.
I think the Slavic autosomal input is probably also, as the papers state, from Slavic women. When the Jews first went to Poland and Lithuania, parts of those areas were still pagan, so the Church wouldn't have been doing any forbidding. Some small part of it may indeed have been the result of rape. I don't know.
That's one of the problems I have with the 780 C.E. date for the admixture. It would have been much easier in an earlier time period. However, there are some interesting documents in the archives of the Bishop of Luni where the Pope sends him a letter saying he has gotten reports that the Bishop was lax in the matter of a Jewish landlord in his area who was still converting his Gentile slaves. I totally understand it. It was Italy then as now. Maybe the Bishop knew him, liked him; the personal is always more important than the legalisms of the state. There was also an interesting document about a Jewish girl who had converted to Catholicism, and to welcome her into the fold, the Bishop sent her a bolt of silk cloth for a wedding dress.
As you can see, in Italy, at least, if a Jewish person did the converting then that Jewish person, male or female, could marry a Gentile. That continued throughout the Middle Ages and beyond. It wasn't "racial" in that sense, as it was in Germany and even in Eastern Europe. Until the Spanish brought it with them to Southern Italy there also wasn't, to the best of my recollection, an organized Inquisition in Italy poking into whether or not somebody who had a Jewish grandmother who converted still seemed to have a disdain for pork or all that nonsense, not until Mussolini, who in an attempt to curry favor with Hitler, had the "Racial Laws" passed. Just more proof fascism, which unfortunately was invented by an Italian, was not "racist" in its original form. The Jews were, however, at various times expelled from different Italian cities, as they were famously expelled from England, for example, and France. It wasn't just Spain. It happened in my own area, in the towns of Pontremoli and Bagnone, in both cases at the instigation of the local merchants, who didn't like the competition. So typical; often rooted in jealousy or resentment.