Surgical video understanding is essential for intraoperative assistance such as workflow monitoring, instrument preparation and procedural documentation. Delivering such assistance in real time requires a model to jointly perceive the ongoing scene, remember past surgical events, and respond before clinically relevant moments pass. Existing approaches provide limited support for jointly retaining procedural evidence and deciding when to respond in streaming surgical video. In this work, we propose SurgStream, a workflow-anchored proactive streaming framework for real-time surgical video understanding. We first introduce Workflow-Anchored Memory (WAM) to record structured instrument-action-target observations and fold them into phase-level summaries, providing compact and queryable surgical memory. Deadline-aware Transition Learning (DTL) is then devised to supervise response states within clinical windows, enabling timely responses before procedural deadlines. We further introduce SurgStream-Bench, comprising 11.7K streaming interactions across various tasks covering real-time perception, backward memory queries, forward proactive QA, proactive alerts and caption memory. Experiments on SurgStream-Bench show gains over state-of-the-art methods, especially in memory and forward proactive QA. The source code will be available.

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