  We present tilting rates for galaxies comparable to the Milky Way (MW) in a
$\Lambda$ cold dark matter cosmological hydrodynamical simulation, and compare
these with the predicted tilting rate detection limit of the {\it Gaia}
satellite $0.28\degrees$Gyr$^{-1}$. We first identify galaxies with mass
comparable to the MW ($9 \times 10^{11} \le M_{200} \le 1.2 \times 10^{12}
\Msun $) and consider the tilting rates between $z=0.3$ and $0$. This sample
yields a tilting rate of $7.6\degrees \pm 4.5\degrees$Gyr$^{-1}$. We constrain
our sample further to exclude any galaxies that have high stellar accretion
during the same time. We still find significant tilting, with an average rate
of $6.3\degrees$Gyr$^{-1}$. Both subsamples tilt with rates significantly above
{\it Gaia}'s predicted detection limit. We show that our sample of galaxies
covers a wide range of environments, including some similar to the MW's. We
find galaxies in denser regions tilt with higher rates then galaxies in less
dense regions. We also find correlations between the angular misalignment of
the hot gas corona, and the tilting rate. {\it Gaia} is likely to be able to
directly measure tilting in the MW. Such a detection will provide an important
constraint on the environment of the MW, including the rate of gas cooling onto
the disc, the shape and orientation of its dark matter halo, and the mass of
the Large Magellanic Cloud. Conversely, failure to detect tilting may suggest
the MW is in a very quiet configuration.
