  For $n<\omega$, let $N_n$ be the minimal iterable proper class mouse $M$ such
that $M\models$ "there are ordinals
$\delta_0<\kappa_0<\ldots<\delta_{n-1}<\kappa_{n-1}$ such that each $\delta_i$
is a Woodin cardinal and each $\kappa_i$ is a strong cardinal", and let
$N_\omega$ be likewise, but with $M\models$ "there is an ordinal $\lambda$
which is a limit of Woodin cardinals and a limit of strong cardinals". Under
appropriate large cardinal hypotheses, Sargsyan and Schindler introduced and
analysed in "Varsovian models I" the Varsovian model of $N_1$, and Sargsyan,
Schindler and the author introduced and analysed in "Varsovian models II" the
Varsovian model of $N_2$. We extend this to $N_\omega$, assuming that
$*$-translation integrates routinely with the P-constructions of this paper
(the write-up of which is yet to be completed). We show, under this assumption,
that $N_\omega$ has a proper class inner model $\mathscr{V}_\omega$ which is a
fully iterable strategy mouse with $\omega$ Woodin cardinals, closed under its
strategy, and that the universe of $\mathscr{V}_\omega$ is the eventual generic
HOD, and the mantle, of $N_\omega$. We also show, under the same assumption,
that the core model $K$ of $N_\omega$ (which can be defined in a natural
manner) is an iterate of $N_\omega$, is an inner model of $\mathscr{V}_\omega$,
and is fully iterable in $M$ and in $\mathscr{V}_\omega$.
