  Eikonal quasinormal modes (QNMs) of black holes (BHs) and parameters of null
geodesics, ultimately tied to the appearance of BHs to external observers, are
known to be related, and the eikonal QNM-BH shadow radii correspondence has
been extensively studied for spherically symmetric BHs. The extension to
rotating BHs is non-trivial, and has been worked out only for equatorial
($m=\pm\ell$) QNMs, or for general modes but limited to the Kerr metric. We
extend the QNM-shadow radius correspondence to more general rotating
space-times, and argue that the requirements for it to hold amount to
conditions on the separability of the Hamilton-Jacobi equation for null
geodesics and the Klein-Gordon equation. Metrics obtained by the Newman-Janis
algorithm enjoy these conditions, provided certain mathematical requirements
are imposed on the line element. We explicitly verify the correspondence for
the rotating Bardeen and Hayward regular BHs, both of which satisfy the
separability requirements. Our findings show that the QNM-shadow radius
correspondence holds for a wide range of axisymmetric space-times beyond Kerr.
This paves the way to potential strong-field multi-messenger tests of
fundamental physics by hearing (via gravitational wave spectroscopy - the
``thunder'') and seeing (via VLBI imaging - the ``lightning'') BHs, although
substantial improvements relative to the current observational sensitivity are
required to make this possible.
