  Dipole-dipole interactions lead to frequency shifts that are expected to
limit the performance of next-generation atomic clocks. In this work, we
compute dipolar frequency shifts accounting for the intrinsic atomic multilevel
structure in standard Ramsey spectroscopy. When interrogating the transitions
featuring the smallest Clebsch-Gordan coefficients, we find that a simplified
two-level treatment becomes inappropriate, even in the presence of large Zeeman
shifts. For these cases, we show a net suppression of dipolar frequency shifts
and the emergence of dominant non-classical effects for experimentally relevant
parameters. Our findings are pertinent to current generations of optical
lattice and optical tweezer clocks, opening a way to further increase their
current accuracy, and thus their potential to probe fundamental and many-body
physics.
