The LHCb Upgrade I trigger system provides full upfront charged-particle reconstruction at the $30\,\mathrm{MHz}$ bunch-crossing rate, enabling, for the first time, the deployment of exclusive selections directly at the first trigger stage (HLT1). A novel real-time exclusive trigger strategy for selecting the $\tau^-\to\phi\mu^-$ decay during Run 3 (2022-2026) proton-proton data taking is presented, together with the first analysis framework developed for this search at a hadron collider. This exclusive selection doubles the signal-event yield with respect to other general-purpose HLT1 selections in Run 3, while the total signal-selection efficiency is increased by about a factor of six relative to Run 2 (2015-2018). The selection algorithm yields an output rate of $1.7\,\mathrm{kHz}$, corresponding to about $0.1\%$ of the total HLT1 output rate, and has operated stably since the start of the 2025 data taking period. As the first exclusive HLT1 trigger for a $\tau$ rare-decay channel at LHCb, it establishes a proof of concept for future decay-specific real-time selections at the high-intensity frontier.