Multi-Agent Path Finding (MAPF) seeks collision-free paths for multiple agents from their respective starting locations to their respective goal locations while minimizing path costs. Although many MAPF algorithms were developed, most of them rely on a common assumption on synchronized actions, where the actions of all agents start at the same time and always take a time unit. This assumption may limit the use of MAPF planners in practice. To get rid of this assumption, recently, an algorithm called Loosely Synchronized Rule-based Planning (LSRP) was proposed, which can find sub-optimal solutions for many agents. However, LSRP often finds poor quality solutions due to its unbounded sub-optimality. This paper develops a new anytime planner called LSRP* that can keep improving solution quality after the initial solution is obtained until the runtime budget depletes. We analyze the properties of LSRP* and test it against several baselines with up to 1000 agents on various maps. LSRP* can handle up to 25% more agents than LSRP and can reduce up to 40% of the solution cost found by LSRP.

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