While recent Spoken Language Models (SLMs) have been actively deployed in real-world scenarios, they lack the capability to discern Third-Party Interruptions (TPI) from the primary user's ongoing flow, leaving them vulnerable to contextual failures. To bridge this gap, we introduce TPI-Train, a dataset of 88K instances designed with speaker-aware hard negatives to enforce acoustic cue prioritization for interruption handling, and TPI-Bench, a comprehensive evaluation framework designed to rigorously measure the interruption-handling strategy and precise speaker discrimination in deceptive contexts. Experiments demonstrate that our dataset design mitigates semantic shortcut learning-a critical pitfall where models exploit semantic context while neglecting acoustic signals essential for discerning speaker changes. We believe our work establishes a foundational resource for overcoming text-dominated unimodal reliance in SLMs, paving the way for more robust multi-party spoken interaction. The code for the framework is publicly available at https://tpi-va.github.io