WASP-12b Exhibits Persistent In-Transit H$\alpha$ Absorption
High-resolution H$\alpha$ and He 10830 {\AA} transit spectroscopy can trace the extended upper atmospheres of exoplanets, typically revealing excess in-transit absorption that occurs in the planet's rest frame. However, hydrodynamical simulations predict that absorption should not strictly follow the planet's orbital motion if gas quickly escapes the planetary Roche lobe and shears into tidal tails. The ultra-hot Jupiter WASP-12b fills an exceptionally large fraction of its Roche lobe ($R_p/R_\mathrm{Roche} = 0.55$), and in this paper we present evidence for tidally-modulated outflow kinematics on this unique planet. We observed a transit of WASP-12b with MMT/Hectochelle and found an excess H$\alpha$ absorption of $3.29^{+0.35}_{-0.36}\%$ (9.3$\sigma$) during the transit and extending at least 30~min post-egress. The signal does not follow the planet's orbital motion and instead appears largely in the stellar rest frame, similar to the extended helium outflows in HAT-P-32b and HAT-P-67b, and is slightly redshifted ($+5.4_{-1.7}^{+1.5}$ km s$^{-1}$). We demonstrate that a similar signal appears across previous high-sensitivity observations dating back to 2012 once the atypical signal kinematics and baseline self-subtraction are taken into account. Finally, we present a three-dimensional Athena++ model of WASP-12b's outflowing atmosphere that simultaneously exhibits the observed extended morphology and lack of planet-aligned velocity gradient. Our study highlights the necessity of three-dimensional simulations for interpreting absorption lines formed in strong outflows, where gas may not simply follow the planet's Doppler signature.