Asynchrony widely exists in microgrids (MGs), such as nonidentical sampling rates and communication delays, which challenges the MG control. This article addresses the asynchronous distributed power control problem of hybrid microgrids, considering different kinds of asynchrony, such as nonidentical sampling rates, and random time delays. To this end, we first formulate the economic dispatch problem of MGs, and devise a synchronous algorithm. Then, we analyze the impact of asynchrony, and propose an asynchronous iteration algorithm based on the synchronous version. By introducing a random clock at each iteration, different types of asynchrony are fitted into a unified framework, where the asynchronous algorithm is converted into a fixed-point iteration problem with a nonexpansive operator, leading to a convergence proof. We further provide an upper bound estimation of the time delay. Moreover, the real-time implementation of the proposed algorithm in both ac and dc MGs is introduced. By measuring the frequency/voltage, the controller is simplified by reducing one order, and adapting to the fast varying load demand. Finally, simulations on a benchmark MG, and experiments are utilized to verify the effectiveness, and advantages of the proposed algorithm.