  We test the consistency of several independent astrophysical measurements of
fundamental dimensionless constants. In particular, we compare direct
measurements of the fine-structure constant $\alpha$ and the proton-to-electron
mass ratio $\mu=m_p/m_e$ (mostly in the optical/ultraviolet) with combined
measurements of $\alpha$, $\mu$ and the proton gyromagnetic ratio $g_p$ (mostly
in the radio band). We point out some apparent inconsistencies, which suggest
that hidden systematics may be affecting some of the measurements. These
findings demonstrate the importance of future more precise measurements with
ALMA, ESPRESSO and ELT-HIRES. We also highlight some of the implications of the
currently available measurements for fundamental physics, specifically for
unification scenarios.
