As large language models (LLMs) continue to grow in size, distributed inference has become increasingly important. Model-parallel strategies must now efficiently scale not only across multiple GPUs but also across multiple nodes. In this work, we present a detailed performance study of multi-node distributed inference using LLMs on GPU-based supercomputers. We conduct experiments with several state-of-the-art inference engines alongside YALIS, a research-oriented prototype engine designed for controlled experimentation. We analyze the strong-scaling behavior of different model-parallel schemes and identify key bottlenecks. Because all-reduce operations are a common performance bottleneck, we develop NVRAR, a hierarchical all-reduce algorithm based on recursive doubling with NVSHMEM. NVRAR achieves up to 1.9$\times$-3.6$\times$ lower latency than NCCL for message sizes between 128 KB and 2 MB on HPE Slingshot and InfiniBand interconnects. Integrated into YALIS, NVRAR achieves up to a 1.72$\times$ reduction in end-to-end batch latency for the Llama 3.1 405B model in multi-node decode-heavy workloads using tensor parallelism.