We present a theoretical and computational study of thixotropic yield-stress materials in cylindrical Couette flows using a novel fluidity-based constitutive model introduced by de Souza Mendes et al. [J. Nonnewtion. Fluid Mech. 261, 1-8 (2018)]. The model relies on measurable rheological properties to couple the equations of motion with an additional equation for the evolution of the material fluidity (i.e., the reciprocal of viscosity). The fluidity itself is used as a structure parameter to assess the material structuring state without the introduction of phenomenological functions or additional parameters. Our simulations parallel rheological tests with a stress-controlled rheometer and are carried out with the material properties obtained experimentally for the laponite suspension from which the model was originally developed. The results reveal that the processes of breakdown and buildup of the microstructure as well as the position of the yield surface in the flow essentially depend on the applied stress and on two material properties associated with distinct thixotropic time scales, namely, the avalanche time and the construction time. The model predictions also capture many features observed in the flow of yield-stress materials with thixotropy, such as the avalanche effect and transient shear banding. We also show that the steady-state flow is uniquely determined by the imposed stress and does not depend on the material initial structuring state. This contrasts with previous reports for nonthixotropic elastoviscoplastic materials, suggesting that nonunique steady flows of structured materials are probably associated with the transient evolution of elastic stresses from a given initial condition.