Non-determinism in a concurrent or distributed setting may lead to many different runs or executions of a program. This paper presents a method to reproduce a specific run for non-deterministic actor or active object systems. The method is based on recording traces of events reflecting local transitions at so-called stable states during execution; i.e., states in which local execution depends on interaction with the environment. The paper formalizes trace recording and replay for a basic active object language, to show that such local traces suffice to obtain global reproducibility of runs; during replay different objects may operate fairly independently of each other and in parallel, yet a program under replay has guaranteed deterministic outcome. We then show that the method extends to the other forms of non-determinism as found in richer active object languages. Following the proposed method, we have implemented a tool to record and replay runs, and to visualize the communication and scheduling decisions of a recorded run, for Real-Time ABS, a formally defined, rich active object language for modeling timed, resource-aware behavior in distributed systems.