This paper investigates the problem of asynchronous repetitive control for a class of discrete-time Markovian switching systems. The control goal is to track a given periodic reference without steady-state error. To achieve this goal, an asynchronous repetitive controller that renders the overall closed-loop switched system mean square stable is proposed. To reflect realistic scenarios, the proposed approach does not assume that the system modes are available synchronously to the controller but instead designs a detector that provides estimated values of the system modes to the controller. Based on a detected-mode-dependent estimator, the plant and asynchronous repetitive controller are formulated as a closed-loop stochastic system. By utilizing tools from stochastic LyapunovKrasovskii stability theory, we develop sufficient conditions in terms of linear matrix inequalities (LMIs) such that the closed-loop system is mean square stable and also simultaneously establish a synthesis procedure for obtaining the gain matrices. We provide numerical simulations on an electrical circuit switched system to illustrate the approach.