  We introduce an extension of the ELVIS project to account for the effects of
the Milky Way galaxy on its subhalo population. Our simulation suite, Phat
ELVIS, consists of twelve high-resolution cosmological dark matter-only (DMO)
zoom simulations of Milky Way-size $\Lambda$CDM~ haloes ($M_{\rm v} = 0.7-2
\times 10^{12} \,\mathrm{M}_\odot$) along with twelve re-runs with embedded
galaxy potentials grown to match the observed Milky Way disk and bulge today.
The central galaxy potential destroys subhalos on orbits with small pericenters
in every halo, regardless of the ratio of galaxy mass to halo mass. This has
several important implications. 1) Most of the $\mathtt{Disk}$ runs have no
subhaloes larger than $V_{\rm max} = 4.5$ km s$^{-1}$ within $20$ kpc and a
significant lack of substructure going back $\sim 8$ Gyr, suggesting that local
stream-heating signals from dark substructure will be rare. 2) The pericenter
distributions of Milky Way satellites derived from $\mathit{Gaia}$ data are
remarkably similar to the pericenter distributions of subhaloes in the
$\mathtt{Disk}$ runs, while the DMO runs drastically over-predict galaxies with
pericenters smaller than 20 kpc. 3) The enhanced destruction produces a tension
opposite to that of the classic `missing satellites' problem: in order to
account for ultra-faint galaxies known within $30$ kpc of the Galaxy, we must
populate haloes with $V_\mathrm{peak} \simeq 7$ km s$^{-1}$ ($M \simeq 3 \times
10^{7} \,\mathrm{M}_\odot$ at infall), well below the atomic cooling limit of
$V_\mathrm{peak} \simeq 16$ km s$^{-1}$ ($M \simeq 5 \times 10^{8}
\,\mathrm{M}_\odot$ at infall). 4) If such tiny haloes do host ultra-faint
dwarfs, this implies the existence of $\sim 1000$ satellite galaxies within 300
kpc of the Milky Way.
