We introduce a combination method à la Nelson-Oppen to solve the satisfiability problem modulo a non-disjoint union of theories connected with bridging functions. The combination method is particularly useful to handle verification conditions involving functions defined over inductive data structures. We investigate the problem of determining the data structure theories for which this combination method is sound and complete. Our completeness proof is based on a rewriting approach where the bridging function is defined as a term rewrite system, and the data structure theory is given by a basic congruence relation. Our contribution is to introduce a class of data structure theories that are combinable with a disjoint target theory via an inductively defined bridging function. This class includes the theory of equality, the theory of absolutely free data structures, and all the theories in between. Hence, our non-disjoint combination method applies to many classical data structure theories admitting a rewrite-based satisfiability procedure.