  The interaction between the supersonic motion of the Large Magellanic Cloud
(LMC) and the Circumgalactic Medium (CGM) is expected to result in a bow shock
that leads the LMC's gaseous disk. In this letter, we use hydrodynamic
simulations of the LMC's recent infall to predict the extent of this shock and
its effect on the Milky Way's (MW) CGM. The simulations clearly predict the
existence of an asymmetric shock with a present day stand-off radius of
$\sim6.7$ kpc and a transverse diameter of $\sim30$ kpc. Over the past 500 Myr,
$\sim8\%$ of the MW's CGM in the southern hemisphere should have interacted
with the shock front. This interaction may have had the effect of smoothing
over inhomogeneities and increasing mixing in the MW CGM. We find observational
evidence of the existence of the bow shock in recent $H\alpha$ maps of the LMC,
providing a potential explanation for the envelope of ionized gas surrounding
the LMC. Furthermore, the interaction of the bow shock with the MW CGM may also
explain observations of ionized gas surrounding the Magellanic Stream. Using
recent orbital histories of MW satellites, we find that many satellites have
likely interacted with the LMC shock. Additionally, the dwarf galaxy Ret2 is
currently sitting inside the shock, which may impact the interpretation of
reported gamma ray excess in Ret2. This work highlights bow shocks associated
with infalling satellites are an under-explored, yet potentially very important
dynamical mixing process in the circumgalactic and intracluster media.
