  The optical depth $\tau$ is the least well determined parameter in the
standard model of cosmology, and one whose precise value is important for both
understanding reionization and for inferring fundamental physics from
cosmological measurements. We forecast how well future epoch of reionization
experiments could constraint $\tau$ using a symmetries-based bias expansion
that highlights the special role played by anisotropies in the power spectrum
on large scales. Given a parametric model for the ionization evolution inspired
by the physical behavior of more detailed reionization simulations, we find
that future 21cm experiments could place tight constraints on the timing and
duration of reionization and hence constraints on $\tau$ that are competitive
with proposed, space-based CMB missions provided they can measure $k\approx
0.1\,h\,\text{Mpc}^{-1}$ with a clean foreground wedge across redshifts
spanning the most active periods of reionization, corresponding to ionization
fractions $0.2 \lesssim x \lesssim 0.8$. Significantly improving upon existing
CMB-based measurements with next-generation 21cm surveys would require
substantially longer observations ($\sim5$ years) than standard
$\mathcal{O}(1000 \,\,\text{hour})$ integration times. Precise measurements of
smaller scales will not improve constraints on $\tau$ until a better
understanding of the astrophysics of reionization is achieved. In the presence
of noise and foregrounds even future 21cm experiments will struggle to
constrain $\tau$ if the ionization evolution deviates significantly from simple
parametric forms.
