  The search for primordial gravitational waves through the $B$-mode
polarization pattern in the CMB is one of the major goals of current and future
CMB experiments. Besides foregrounds, a potential hurdle in this search is the
anisotropic secondary $B$-mode polarization generated by the scattering of CMB
photons off free electrons produced during patchy cosmological reionization.
Robust predictions of these secondary anisotropies are challenging because of
uncertainties in the reionization history. In this paper, we revise estimates
of the reionization-induced $B$-mode signal by incorporating recent advances in
the understanding of reionization through observations of the Lyman-$\alpha$
forest. To derive these $B$-mode estimates, we use high-dynamic-range radiative
transfer simulations of reionization that are calibrated to the Ly$\alpha$
data. These simulations are also consistent with a variety of other
high-redshift observations. We find that around multipoles $\ell\approx 100$,
reionization induces $B$-mode power with $\ell(\ell+1)C_\ell^{BB}/2\pi\approx
4\times 10^{-6}\,\mu$K$^2$. This secondary signal is thus at the level of the
primordial signal with the tensor-to-scalar ratio $r<10^{-4}$, and can increase
by a factor of $\sim 50$ if reionization is sourced by highly clustered sources
residing in haloes with mass of $\sim 10^{11}$ M$_\odot$. Our findings suggest
that the contribution of patchy reionization to the search for primordial
gravitational waves is unlikely to be a concern for currently planned CMB
experiments.
