Abstract We show that tableau methods for satisfiability in non-classical logics can be supported naturally in SMT solving via the framework of user-propagators. By way of demonstration, we implement the description logic $$\mathcal {ALC}$$ in the Z3 SMT solver and show that working with user-propagators allows us to significantly outperform encodings to first-order logic with relatively little effort. We promote user-propagators for creating solvers for non-classical logics based on tableau calculi.