  We present the first simulations of the high-redshift Ly$\alpha$ intensity
field that account for scattering in the intergalactic medium (IGM). Using a 3D
Monte Carlo radiative transfer code, we find that Ly$\alpha$ scattering smooths
spatial fluctuations in the Ly$\alpha$ intensity on small scales and that the
spatial dependence of this smoothing depends strongly on the mean neutral
fraction of the IGM. Our simulations find a strong effect of reionization on
$k=0.1-1~{\rm Mpc^{-1}}$, with $P_{\rm Ly\alpha}\propto k^{-1.75}$ for
$\bar{x}_{\rm HI} = 0.63$ and $P_{\rm Ly\alpha} \propto k^{-2.2}$ for
$\bar{x}_{\rm HI} = 0.86$ in contrast to $P_{\rm Ly\alpha}\propto k^{-1.5}$
after reionization. At wavenumbers of $k>1 ~ {\rm Mpc^{-1}}$, we find that the
signal is sensitive to the emergent Ly$\alpha$ line profiles from galaxies. We
also demonstrate that the cross-correlation between a Ly$\alpha$ intensity map
and a future galaxy redshift survey could be detected on large scales by an
instrument similar to SPHEREx, and over a wide range of scales by a
hypothetical intensity mapping instrument in the vein of CDIM.
