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We examine the rheology of bicontinuous interfacially jammed emulsion gels (bijels) beyond the limit of linear viscoelasticity and propose a simple model that connects bijel rheology to anticipated microstructural changes in the continuous particle-laden interface. The basic elements of our model are constructed from a linear stress relaxation experiment, which suggests glasslike rheology mediated by α (out-of-cage) and β (in-cage) particle dynamics along the interface in this limit. Extending to medium amplitude oscillatory shear experiments then reveals signatures of nonlinearity, which we rationalize as the combined effect of shear-induced dilation of the interface and its simultaneous recompaction by interfacial tension, as well as potential buckling along portions of the interface under compression. Informed by these observations, we present a double Maxwell model with sigmoidal nonlinearities introduced to account for how interfacial dilation and recompaction affect the intercage particle dynamics (α relaxation) along the interface and buckling. This simple model successfully captures the general features of nonlinear rheology in bijels, indicating that their linear-to-nonlinear transition is associated with loss of compaction along the dilated and buckling along the compacted regions of the interface. Our results shed initial light on the microstructural origins of nonlinear rheology in bijels and the reconfigurability afforded in these systems by the balance of glassy particle dynamics and interfacial recompaction under shear deformation.