Correct! In a common law union, the couple is not legally married but is eligible for tax benefits similar to those enjoyed by married couples. When it comes to other matters, however, such as the breakdown of the relationship, a common law couple does not have the same status as a married couple and as such, is not subject to the Rabbinate’s jurisdiction. Children who are born to parents in a common law union are not considered mamzerim (as long as the woman is not concurrently halakhically married to another man), and are able to marry whomever they please.
The Rabbinate has no authority over me because we are in a common law union.