  We demonstrate precision measurement and control of inhomogeneous broadening
in a multi-ion clock consisting of three $^{176}$Lu$^+$ ions. Microwave
spectroscopy between hyperfine states in the $^3D_1$ level is used to
characterise differential systematic shifts between ions, most notably those
associated with the electric quadrupole moment. By appropriate alignment of the
magnetic field, we demonstrate suppression of these effects to the $\sim
10^{-17}$ level relative to the $^1S_0\leftrightarrow{}^3D_1$ optical
transition frequency. Correlation spectroscopy on the optical transition
demonstrates the feasibility of a 10s Ramsey interrogation in the three ion
configuration with a corresponding projection noise limited stability of
$\sigma(\tau)=8.2\times 10^{-17}/\sqrt{\tau}$
