  The outflows of oxygen-rich asymptotic giant branch (AGB) stars are thought
to be driven by radiation pressure by photon scattering on grains with sizes of
tenths of microns. The details of the formation of dust in the extended
atmospheres of these stars and the mass-loss process is still not well
understood. We obtained quasi-simultaneous observations of the AGB star Mira
using ALMA and ZIMPOL to probe the distribution of gas and large dust grains,
respectively. The polarized light images show dust around Mira~A, the companion
(Mira~B) and in a trail that connects the two sources. ALMA reveals that dust
around Mira~A is contained in a high-gas-density region with a significant
fraction of the polarized light arising from its edge. We constrained the gas
density, temperature, and velocity within a few stellar radii from the star by
modelling the CO $v=1, J=3-2$ line. We find a mass $(\sim 3.8 \pm 1.3) \times
10^{-4}~M_\odot$ to be contained between the stellar millimetre photosphere,
$R^{\rm 338~GHz}_\star$, and $4~R^{\rm 338~GHz}_\star$. Our best-fit models
with lower masses also reproduce the $^{13}$CO $v=0, J=3-2$ line emission from
this region. We find TiO$_2$ and AlO abundances corresponding to 4.5% and $<
0.1$% of the total titanium and aluminium expected for a solar-composition gas.
The low abundance of AlO allows for efficient Al depletion into dust already
very close to the star, as expected from thermal dust emission observations and
theoretical calculations of Mira variables. We constrain the presence of
aluminium oxide grains based on the scattered light observations and our
gas-phase model. We find that aluminium oxide grains can account for a
significant fraction of the total aluminium atoms in this region only if the
grains have sizes $\lesssim 0.02~\mu$m. This is an order of magnitude smaller
than the maximum sizes predicted by dust-formation and wind-driving models.
