As is well-known, every real (and complex) hyperplane arrangement divides the space into a set of faces, on which one can define a natural product, inducing a structure of a left regular band on this set. One can ask whether the same is possible for other arrangements of hypersurfaces. In this paper, we try to answer this question for the simplest generalization of hyperplane arrangements, that is, conic--line arrangements in the plane. Investigating the different algebraic structures induced on the face poset of a conic--line arrangement, we present two possibilities for 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.