  Cosmic birefringence is predicted if an axion-like particle (ALP) moves after
the recombination. We show that this naturally happens if the ALP is coupled to
the dark matter density because it then acquires a large effective mass after
the matter-radiation equality. Our scenario applies to a broad range of the ALP
mass $m_\phi \lesssim 10^{-28}$ eV, even smaller than the present Hubble
constant. We give a simple model to realize this scenario, where dark matter is
made of hidden monopoles, which give the ALP such a large effective mass
through the Witten effect. The mechanism works if the ALP decay constant is of
order the GUT scale without a fine-tuning of the initial misalignment angle.
For smaller decay constant, the hidden monopole can be a fraction of dark
matter. We also study the implications for the QCD axion, and show that the
domain wall problem can be solved by the effective mass.
