We consider a quasilinear elliptic transmission problem where the nonlinearity changes discontinuously across two subdomains. By a reformulation of the problem via a Kirchhoff transformation, we first obtain linear problems on the subdomains together with nonlinear transmission conditions and then a nonlinear Steklov–Poincaré interface equation. We introduce a Dirichlet–Neumann iteration for this problem and prove convergence to a unique solution in one space dimension. Finally we present numerical results in two space dimensions suggesting that the algorithm can be applied successfully in more general cases.