The PGAS model is well suited for executing irregular applications on cluster-based systems, due to its efficient support for short, one-sided messages. However, there are currently two major limitations faced by PGAS applications. The first relates to scalability: despite the availability of APIs that support non-blocking operations in special cases, many PGAS operations on remote locations are synchronous by default, which can lead to long-latency stalls and poor scalability. The second relates to productivity: while it is simpler for the developer to express all communications at a fine-grained granularity that is natural to the application, experiments have shown that such a natural expression results in performance that is 20x slower than more efficient but less productive code that requires manual message aggregation and termination detection. In this paper, we introduce a new programming system for PGAS applications, in which point-to-point remote operations can be expressed as fine-grained asynchronous actor messages. In this approach, the programmer does not need to worry about programming complexities related to message aggregation and termination detection. Our approach can also be viewed as extending the classical Bulk Synchronous Parallelism model with fine-grained asynchronous communications within a phase or superstep. We believe that our approach offers a desirable point in the productivity-performance space for PGAS applications, with more scalable performance and higher productivity relative to past approaches. Specifically, for seven irregular mini-applications from the Bale benchmark suite executed using 2048 cores in the NERSC Cori system, our approach shows geometric mean performance improvements of >=20x relative to standard PGAS versions (UPC and OpenSHMEM) while maintaining comparable productivity to those versions.