Viscoplastic fluids play a central role in various industrial and natural flow applications. When using Bingham-like models, extreme viscosity contrasts—going towards “infinity” in unyielded (plug) regions—pose substantial numerical challenges. Additionally, the common free-surface or two-phase nature of the flow requires appropriate interface treatments, and the fine meshes needed to resolve yield surfaces and fluid interfaces can lead to high computational costs. A promising technique to speed up simulations is implicit-explicit (IMEX) time stepping, which treats stiff terms implicitly while keeping others explicit. In non-Newtonian simulations, extrapolating nonlinear terms allows bypassing costly fixed-point iterations without sacrificing numerical stability or accuracy. In this talk, we show new IMEX methods that also split pressure from velocity, thereby simplifying implementation and reducing computational costs. Our numerical examples include column collapse benchmarks involving Bingham and granular models.