  The axion-photon coupling allows the existence of a magnetically and
electrically charged black hole (BH) solution endowed with a pseudo-scalar
hair. For the Reissner-Nordstrom BH with a given total charge and mass, it is
known that the quasinormal modes (QNMs) are independent of the mixture between
the magnetic and electric charges due to the presence of electric-magnetic
duality. We show that the BH with an axion hair breaks this degeneracy by
realizing nontrivial QNMs that depend on the ratio between the magnetic and
total charges. Thus, the upcoming observations of BH QNMs through gravitational
waves offer an exciting possibility for probing the existence of both magnetic
monopoles and the axion coupled to photons.
