Large language models (LLMs) are rapidly transitioning into medical clinical use, yet their reliability under realistic, multi-turn interactions remains poorly understood. Existing evaluation frameworks typically assess single-turn question answering under idealized conditions, overlooking the complexities of medical consultations where conflicting input, misleading context, and authority influence are common. We introduce MedQA-Followup, a framework for systematically evaluating multi-turn robustness in medical question answering. Our approach distinguishes between shallow robustness (resisting misleading initial context) and deep robustness (maintaining accuracy when answers are challenged across turns),while also introducing an indirect–direct axis that separates contextual framing(indirect) from explicit suggestion (direct). Using controlled interventions on theMedQA dataset, we evaluate five state-of-the-art LLMs and find that while models perform reasonably well under shallow perturbations, they exhibit severe vulnerabilities in multi-turn settings, with accuracy dropping from 91.2% to as low as13.5% for Claude Sonnet 4. Counterintuitively, indirect, context-based interventions are often more harmful than direct suggestions, yielding larger accuracy drops across models and exposing a significant vulnerability for clinical deployment. Further compounding analyses reveal model differences, with some showing further performance drops under repeated interventions while others partially recovering or even improving. These findings highlight multi-turn robustness as a critical but underexplored dimension for safe and reliable deployment of medical LLMs.Dataset and code available on HuggingFace and GitHub.