An arrangement of curves in the real plane divides it into a collection of faces. Already in the case of line arrangements, this collection can be given a structure of a left regular band and one can ask whether the same is possible for other arrangements. In this paper, we try to answer this question for the simplest generalization of line arrangements, that is, conic--line arrangements. Investigating the different algebraic structures induced on the face poset of a conic--line arrangement, we present two possibilities for generalizing the product and its associated structures. We also study the structure of sub left regular bands induced by these arrangements. We finish with some combinatorial properties of conic--line arrangements.