We test the consistency of several independent astrophysical measurements of fundamental dimensionless constants. In particular, we compare direct measurements of the fine-structure constant α and the proton-to-electron mass ratio µ = mp/me (mostly in the optical/ultraviolet) with combined measurements of α, µ and the proton gyromagnetic ratio gp (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.