  We provide an atomic decomposition of the product Hardy spaces
$H^p(\widetilde{X})$ which were recently developed by Han, Li, and Ward in the
setting of product spaces of homogeneous type $\widetilde{X} = X_1 \times X_2$.
Here each factor $(X_i,d_i,\mu_i)$, for $i = 1$, $2$, is a space of homogeneous
type in the sense of Coifman and Weiss.
  These Hardy spaces make use of the orthogonal wavelet bases of Auscher and
Hyt\"onen and their underlying reference dyadic grids.
  However, no additional assumptions on the quasi-metric or on the doubling
measure for each factor space are made. To carry out this program, we introduce
product $(p,q)$-atoms on $\widetilde{X}$ and product atomic Hardy spaces
$H^{p,q}_{{\rm at}}(\widetilde{X})$. As consequences of the atomic
decomposition of $H^p(\widetilde{X})$, we show that for all $q > 1$ the product
atomic Hardy spaces coincide with the product Hardy spaces, and we show that
the product Hardy spaces are independent of the particular choices of both the
wavelet bases and the reference dyadic grids. Likewise, the product Carleson
measure spaces ${\rm CMO}^p(\widetilde{X})$, the bounded mean oscillation space
${\rm BMO}(\widetilde{X})$, and the vanishing mean oscillation space ${\rm
VMO}(\widetilde{X})$, as defined by Han, Li, and Ward, are also independent of
the particular choices of both wavelets and reference dyadic grids.
