While large language models (LLMs) have shown promise in diagnostic dialogue, their capabilities for effective management reasoning - including disease progression, therapeutic response, and safe medication prescription - remain under-explored. We advance the previously demonstrated diagnostic capabilities of the Articulate Medical Intelligence Explorer (AMIE) through a new LLM-based agentic system optimised for clinical management and dialogue, incorporating reasoning over the evolution of disease and multiple patient visit encounters, response to therapy, and professional competence in medication prescription. To ground its reasoning in authoritative clinical knowledge, AMIE leverages Gemini's long-context capabilities, combining in-context retrieval with structured reasoning to align its output with relevant and up-to-date clinical practice guidelines and drug formularies. In a randomized, blinded virtual Objective Structured Clinical Examination (OSCE) study, AMIE was compared to 21 primary care physicians (PCPs) across 100 multi-visit case scenarios designed to reflect UK NICE Guidance and BMJ Best Practice guidelines. AMIE was non-inferior to PCPs in management reasoning as assessed by specialist physicians and scored better in both preciseness of treatments and investigations, and in its alignment with and grounding of management plans in clinical guidelines. To benchmark medication reasoning, we developed RxQA, a multiple-choice question benchmark derived from two national drug formularies (US, UK) and validated by board-certified pharmacists. While AMIE and PCPs both benefited from the ability to access external drug information, AMIE outperformed PCPs on higher difficulty questions. While further research would be needed before real-world translation, AMIE's strong performance across evaluations marks a significant step towards conversational AI as a tool in disease management.