Maximally inflected real trigonal curves on Hirzebruch surfaces

In 2014 A. Degtyarev, I. Itenberg and the author gave a description, up to fiberwise equivariant deformations, of maximally inflected real trigonal curves of type~I (over a base $B$ of an arbitrary genus) in terms of the combinatorics of sufficiently simple graphs and for $B=\mathbb{P}^1$ obtained a complete classification of such curves. In this paper, the mentioned results are extended to maximally inflected real trigonal curves of type~II over $B=\mathbb{P}^1$. As usual, the results for such curves are extended to real Jacobian elliptic surfaces with all singular fibers real.