  Conventional approaches for lattice dynamics based on static interatomic
forces do not fully account for the effects of time-reversal-symmetry breaking
in magnetic systems. Recent approaches to rectify this involve incorporating
the first-order change in forces with atomic velocities under the assumption of
adiabatic separation of electronic and nuclear degrees of freedom. In this
work, we develop a first-principles method to calculate this velocity-force
coupling in extended solids, and show via the example of ferromagnetic CrI$_3$
that, due to the slow dynamics of the spins in the system, the assumption of
adiabatic separation can result in large errors for splittings of zone-center
chiral modes. We demonstrate that an accurate description of the lattice
dynamics requires treating magnons and phonons on the same footing.
