We present a physically consistent multiport framework for stacked intelligent metasurfaces (SIMs) with linear and explicit nonlinear terminations. The model provides closed-form input--output relations in the linear case and fixed-point forward evaluation in the nonlinear case, with adjoint-based gradients for optimization in both settings. Under stage-isolated SIM structure, complexity remains $\mathcal{O}(QK^3)$. In a 28 GHz near-field localization case study, nonlinear terminations improve transfer-function matching and reduce mean localization error, close to the ideal benchmark.