Ironically, "Messianic Jew" is just a modern description; NOT a denomination or religion.  It also often isn't an accurate name for Messianics - some who are grouped instead by terms such as  Nazarenes, Notsarim, Yisra'elites, Nazarene Israelites, or terms such as Followers of the Way, or Torah Observant Yisra'elites.... the list goes on.  While some observers of Judaism have transitioned to a belief in Messiah ( one of the literal definitions of "Messianic Jew"), the majority of people adopting a Messianic understanding of Faith are NOT "Jews", nor are they attempting to convert to Judaism....

   There is a much deeper truth, along with one of the many ongoing fulfillments of "Old Testament" prophecy that continues to this day:  They represent the descendants of the bulk of the Northern House of Yisra'el, who were sentenced to exile and punishment.  The proscribed period of punishment has recently been completed and those Children are now coming back to resume a covenantal relationship with Yah.