We propose a physical model for the nonlinear inelastic mechanics of sticky biopolymer networks with potential applications to inelastic cell mechanics. It consists of a minimal extension of the glassy wormlike chain (Gwlc) model, which has recently been highly successful as a quantitative mathematical description of the viscoelastic properties of biopolymer networks and cells. To extend its scope to nonequilibrium situations, where the thermodynamic state variables may evolve dynamically, the Gwlc is furnished with an explicit representation of the kinetics of breaking and reforming sticky bonds. In spite of its simplicity, the model exhibits many experimentally established nontrivial features such as power-law rheology, stress stiffening, fluidization and cyclic softening effects.