  We propose a scenario where the spontaneous breakdown of the Peccei-Quinn
symmetry leads to monopole production. Both the axion and the monopole
contribute to dark matter, and the Witten effect on the axion mass is a
built-in feature. In the KSVZ-type axion model, seemingly different vacua are
actually connected by the hidden gauge symmetry, which makes the axionic string
unstable and separate into two Alice strings. The Alice string is attached to a
single domain wall due to the QCD instanton effect, solving the domain-wall
problem. This is in the same spirit of the Lazarides-Shafi mechanism, although
the discrete Peccei-Quinn symmetry is not embedded into the center of the
original gauge symmetry. In the DFSZ-type axion model, the domain-wall problem
is avoided by the Witten effect. If the Peccei-Quinn symmetry is explicitly
broken by a small amount, monopoles acquire a tiny electric charge and become
mini-charged dark matter. Interestingly, the quality of the Peccei-Quinn
symmetry is closely tied to darkness of dark matter.
