  We consider a dark matter (DM) model offering a very natural explanation of
the observed relation, $\Omega_{\rm dark} \sim \Omega_{\rm visible}$. This
generic consequence of the model is a result of the common origin of both types
of matter (DM and visible) which are formed during the same QCD transition. The
masses of both types of matter in this framework are proportional to one and
the same dimensional parameter of the system, $\Lambda_{\rm QCD}$. The focus of
the present work is the detail study of the dynamics of the $\cal{CP}$-odd
coherent axion field $a(x)$ just before the QCD transition. We argue that the
baryon charge separation effect on the largest possible scales inevitably
occurs as a result of merely existence of the coherent axion field in early
Universe. It leads to preferential formation of one species of nuggets on the
scales of the visible Universe where the axion field $a(x)$ is coherent. A
natural outcome of this preferential evolution is that only one type of the
visible baryons remains in the system after the nuggets complete their
formation. This represents a specific mechanism on how the baryon charge
separation mechanism (when the Universe is neutral, but visible part of matter
consists the baryons only) replaces the conventional "baryogenesis" scenarios.
