Jehovah's Witnesses may like to believe they are the "mix of Judaism and Christianity," and I've heard them use this term for themselves before.
But in reality, there are still members of the original Jewish Christian movement that exist. The Jerusalem Church was dissolved in 135 CE when the Romans dismissed all Jews from Jerusalem after crushing the Bar Kokhba rebellion. While it is not clear what exactly happened to Bishop Judas Kyriakos himself who was leading the Church at that time, the congregants eventually got absorbed into other churches via the resulting Diaspora.
Some of my ancestors may have been members of the original Jerusalem Church (or so I've recently learned via a group studying the history of Crypto/Sephardic Jews), and today there are pockets of Jewish Christians around the world, though their numbers are miniscule. The Catholic Church has a sizeable number of these. Known there as "Hebrew Catholics," they observe a practice similar to what is described at Acts 21:20 while remaining fully approved members of the Roman Catholic Church.
"Messianic Jews" aren't in this group, by the way, because they are Christian Fundamentalist Evangelicals who believe that Jews must accept Christ (or "burn in hell," as one of them told me would happen if I didn't "accept Christ as my personal Savior"), and generally advocate adherence to the Mosaic Law even for Gentile Christians. They number more than 60% Gentiles in their membership.
People like the Hebrew Catholics and other similar Jews are the real "mix of Judaism and Christianity," and there is even a parish of Hebrew-speaking Catholics in Israel that lives relatively peacefully among the Jews. I know some other Jewish Christians here in the US very well, and believe me, Watchtower religion doesn't come close at all.