Nonlinear rheological properties are often relevant in understanding the response of a material to its intended environment. For example, many gastropods crawl on a thin layer of pedal mucus using a technique called adhesive locomotion, in which the gel structure is periodically ruptured and reformed. We present a mechanical model that captures the key features of this process and suggests that the most important properties for optimal inclined locomotion are a large, reversible yield stress, followed by a small shear viscosity and a short thixotropic restructuring time. We present detailed rheological measurements of native pedal mucus in both the linear and nonlinear viscoelastic regimes and compare this "rheological fingerprint" with corresponding observations of two bioinspired slime simulants, a polymer gel and a clay-based colloidal gel, that are selected on the basis of their macroscopic rheological similarities to gastropod mucin gels. Adhesive locomotion (of snails or mechanical crawlers) imposes a large-amplitude pulsatile simple shear flow onto the supporting complex fluid, motivating the characterization of nonlinear rheological properties with large amplitude oscillatory shear (LAOS). We represent our results in the form of Lissajous curves of oscillatory stress against time-varying strain. The native pedal mucus gel is found to exhibit a pronounced strain-stiffening response, which is not imitated by either simulant.