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This paper is concerned with robust, tube-based MPC for control systems with bounded time-varying disturbances. In tube MPC, predicted trajectories are replaced by a robust forward invariant tube (RFIT), a set-valued function enclosing all possible state trajectories under a given feedback control law, regardless of the uncertainty realization. In this paper, the main idea is to characterize RFITs with polytopic cross-sections via a min-max differential inequality for their support functions. This result leads to a conservative but tractable polytopic tube MPC formulation, which can be solved using existing optimal control solvers. The corresponding theoretical developments are illustrated by a numerical case study.