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Large language models (LLMs) are increasingly used to support clinical practice. Medical coding is a process where experts map unstructured clinical notes to alphanumeric codes for diagnoses and procedures. We introduce Code Like Humans: a new reasoning-based framework for medical coding with LLMs that implements official coding guidelines for human coders. It is the first solution to support the full US ICD-10 classification space (+70K labels), and it achieves comparable results with fine-tuned classifier models on rare classes. However, the performance on frequent classes is still lacking, and we also find that some medical conditions are systematically undercoded. These limitations suggest that LLM-based medical coding should assist human coders rather than automate the task.