We study the problem of how the dual complex of the special fiber of a strict normal crossings degeneration
$\mathscr{X}_{R}$
changes under products. We view the dual complex as a skeleton inside the Berkovich space associated to
$X_{K}$
. Using the Kato fan, we define a skeleton
$\text{Sk}(\mathscr{X}_{R})$
when the model
$\mathscr{X}_{R}$
is log-regular. We show that if
$\mathscr{X}_{R}$
and
$\mathscr{Y}_{R}$
are log-smooth, and at least one is semistable, then
$\text{Sk}(\mathscr{X}_{R}\times _{R}\mathscr{Y}_{R})\simeq \text{Sk}(\mathscr{X}_{R})\times \text{Sk}(\mathscr{Y}_{R})$
. The essential skeleton
$\text{Sk}(X_{K})$
, defined by Mustaţă and Nicaise, is a birational invariant of
$X_{K}$
and is independent of the choice of
$R$
-model. We extend their definition to pairs, and show that if both
$X_{K}$
and
$Y_{K}$
admit semistable models,
$\text{Sk}(X_{K}\times _{K}Y_{K})\simeq \text{Sk}(X_{K})\times \text{Sk}(Y_{K})$
. As an application, we compute the homeomorphism type of the dual complex of some degenerations of hyper-Kähler varieties. We consider both the case of the Hilbert scheme of a semistable degeneration of K3 surfaces, and the generalized Kummer construction applied to a semistable degeneration of abelian surfaces. In both cases we find that the dual complex of the
$2n$
-dimensional degeneration is homeomorphic to a point,
$n$
-simplex, or
$\mathbb{C}\mathbb{P}^{n}$
, depending on the type of the degeneration.