  We introduce the notion of regularity for a relative holonomic $\mathcal
D$-module in the sense of arXiv:1204.1331. We prove that the solution functor
from the bounded derived category of regular relative holonomic modules to that
of relative constructible complexes is essentially surjective by constructing a
right quasi-inverse functor. When restricted to relative $\mathcal D$-modules
underlying a regular mixed twistor $\mathcal D$-module, this functor satisfies
the left quasi-inverse property.
