Large Language Models (LLMs) often fail to recognize fallacious reasoning in real-world interactions, despite strong performance on static fallacy detection tasks. We define this ability as fallacy awareness, the capacity to autonomously perceive and resist fallacies in dynamic, pragmatic contexts. To study this, we introduce ISFallacy, a large-scale Chinese benchmark of 50K interactive scenarios spanning six fallacy types, five social interaction settings, diverse role relationships, and personality traits. We further propose FATE, a two-stage evaluation framework that assesses fallacy awareness without explicit cues, combining natural dialogue responses and reasoning-based decisions. Experiments on five representative LLMs reveal a substantial gap between fallacy classification and awareness, with models particularly vulnerable to emotion-driven fallacies and scenarios involving cooperative or trust-based relationships. Deeper analysis uncovers a cognition–behavior gap and fragile internal representations underlying awareness failures. Our work establishes a foundation for evaluating and enhancing the robustness of LLMs against fallacious reasoning in interactive settings.