The bulk synchronous parallel (BSP) model struggles with irregular workloads due to rigid global communication. While fine-grained asynchronous BSP (FA-BSP) improves overlap, existing implementations typically rely on a limiting one-process-per-core model. This paper proposes a multithreaded FA-BSP approach combining Lightweight Communication Interface (LCI) and OpenMP to fully exploit multicore architectures. We evaluate this design using the NAS Parallel Benchmark Integer Sort (IS), retaining the original irregular Gaussian distribution to rigorously test load balancing. By replacing synchronous MPI collectives with OpenMP multithreading and LCI's fine-grained, zero-copy active messages, we enable efficient computation-communication overlap. Our evaluation demonstrates that multithreaded FA-BSP significantly outperforms traditional bulk-synchronous MPI implementations, offering a scalable solution for irregular scientific applications.