Implementing correct distributed systems is an error-prone task. Runtime Verification (RV) offers a lightweight formal method to improve reliability by monitoring system executions against correctness properties. However, applying RV in distributed settings - where no process has global knowledge - poses fundamental challenges, particularly under full asynchrony and fault tolerance. This paper addresses the Distributed Runtime Verification (DRV) problem under such conditions. In our model, each process in a distributed monitor receives a fragment of the input word describing system behavior and must decide whether this word belongs to the language representing the correctness property being verified. Hence, the goal is to decide languages in a distributed fault-tolerant manner. We propose several decidability definitions, study the relations among them, and prove possibility and impossibility results. One of our main results is a characterization of the correctness properties that can be decided asynchronously. Remarkably, it applies to any language decidability definition. Intuitively, the characterization is that only properties with no real-time order constraints can be decided in asynchronous fault-tolerant settings. These results expose the expressive limits of DRV in realistic systems, as several properties of practical interest rely on reasoning about real-time order of events in executions. To overcome these limitations, we introduce a weaker model where the system under inspection is verified indirectly. Under this weaker model we define predictive decidability, a decidability definition that turn some real-time sensitive correctness properties verifiable. Our framework unifies and extends existing DRV theory and sharpens the boundary of runtime monitorability under different assumptions.