  Acoustic waves traveling through the early Universe imprint a characteristic
scale in the clustering of galaxies, QSOs and inter-galactic gas. This scale
can be used as a standard ruler to map the expansion history of the Universe, a
technique known as Baryon Acoustic Oscillations (BAO). BAO offer a
high-precision, low-systematics means of constraining our cosmological model.
The statistical power of BAO measurements can be improved if the `smearing' of
the acoustic feature by non-linear structure formation is undone in a process
known as reconstruction. In this paper we use low-order Lagrangian perturbation
theory to study the ability of $21\,$cm experiments to perform reconstruction
and how augmenting these surveys with galaxy redshift surveys at relatively low
number densities can improve performance. We find that the critical number
density which must be achieved in order to benefit $21\,$cm surveys is set by
the linear theory power spectrum near its peak, and corresponds to densities
achievable by upcoming surveys of emission line galaxies such as eBOSS and
DESI. As part of this work we analyze reconstruction within the framework of
Lagrangian perturbation theory with local Lagrangian bias, redshift-space
distortions, ${\bf k}$-dependent noise and anisotropic filtering schemes.
