Consumer products, such as foods, contain numerous polymeric and particulate additives that play critical roles in maintaining their stability, quality and function. The resulting materials exhibit complex bulk and interfacial rheological responses, and often display a distinctive power-law response under standard rheometric deformations. These power laws are not conveniently described using conventional rheological models, without the introduction of a large number of relaxation modes. Large amplitude oscillatory shear (LAOS) is used as a tool to probe the nonlinear rheological response of a model elasto-viscoplastic material (a Carbopol microgel). In contrast to most recent studies, these large amplitude measurements are carried out in a stress-controlled manner. We outline a descriptive framework of characterization measures for nonlinear rheology under stress-controlled LAOS, and this is contrasted experimentally to the strain-controlled framework that is more commonly used. We show that this stress-controlled methodology allows for a physically intuitive interpretation of the yielding behavior of elasto-viscoplastic materials. The insight gained into the material behavior through these nonlinearmeasures is then used to develop two constitutive models that prescribe the rheological response of the Carbopol microgel. We show that these two successively more sophisticated constitutive models, which are based on the idea of strain decomposition, capture in a compact manner the important features of the nonlinear rheology of the microgel. The second constitutive model, which incorporates the concept of kinematic hardening, embodies all of the essential behaviors exhibited by Carbopol. These include elasto-viscoplastic creep and time-dependent viscosity plateaus below a critical stress, a viscosity bifurcation at the critical stress, and Herschel–Bulkley flow behavior at large stresses.