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In medical coding, experts map unstructured clinical notes to alphanumeric codes for diagnoses and procedures. We introduce ‘Code Like Humans’: a new agentic framework for medical coding with large language models. It implements official coding guidelines for human experts, and it is the first solution that can support the full ICD-10 coding system (+70K labels). It achieves the best performance to date on rare diagnosis codes. Fine-tuned discriminative classifiers retain an advantage for high-frequency codes, to which they are limited. Towards future work, we also contribute an analysis of system performance and identify its ‘blind spots’ (codes that are systematically undercoded).