The standard lore on global symmetries is that they cannot be exact in theoretical frameworks where gravity is properly accounted for. In this talk, we discuss how this statement can be observationally investigated using gravitational waves. We focus on the axion solution to the strong CP problem, which in its standard implementation exhibits delicate sensitivity to generic Peccei-Quinn breaking contributions.  Heavy QCD axion models are appealing because they avoid this so-called “quality problem”. We discuss how generic realizations of this framework can be probed by the LIGO-Virgo-KAGRA interferometers, through the stochastic gravitational wave (GW) signal sourced by the long-lived axionic string-domain wall network, and by upcoming measurements of the neutron Electric Dipole Moment. 

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