Medical coding translates free-text clinical documentation into standardized codes drawn from classification systems that contain tens of thousands of entries and are updated annually. It is central to billing, clinical research, and quality reporting, yet remains largely manual, slow, and error-prone. Existing automated approaches learn to predict a fixed set of codes from labeled data, thereby preventing adaptation to new codes or different coding systems without retraining on different data. They also provide no explanation for their predictions, limiting trust in safety-critical settings. We introduce Symphony for Medical Coding, a system that approaches the task the way expert human coders do: by reasoning over the clinical narrative with direct access to the coding guidelines. This design allows Symphony to operate across any coding system and to provide span-level evidence linking each predicted code to the text that supports it. We evaluate on two public benchmarks and three real-world datasets spanning inpatient, outpatient, emergency, and subspecialty settings across the United States and the United Kingdom. Symphony achieves state-of-the-art results across all settings, establishing itself as a flexible, deployment-ready foundation for automated clinical coding.