  Given a Galois cover $Y \to X$ of smooth projective geometrically connected
curves over a complete discrete valuation field $K$ with algebraically closed
residue field, we define a semistable model of $Y$ over the ring of integers of
a finite extension of $K$, which we call the relatively stable model
$\mathcal{Y}^{\mathrm{rst}}$ of $Y$, and we discuss its properties. We focus on
the case when $Y : y^2 = f(x)$ is a hyperelliptic curve, viewed as a degree-$2$
cover of the projective line $X := \mathbb{P}_K^1$, and demonstrate a practical
way to compute the relatively stable model. In the case of residue
characteristic $p \neq 2$, the components of the special fiber
$(\mathcal{Y}^{\mathrm{rst}})_s$ correspond precisely to the non-singleton
clusters of roots of the defining polynomial $f$, i.e. the subsets of roots of
$f$ which are closer to each other than to the other roots of $f$ with respect
to the induced discrete valuation on the splitting field; this relationship,
however, is far less straightforward in the $p=2$ case, which is our main focus
(the techniques we introduce nevertheless also allow us to recover the simpler,
already-known results in the $p\neq 2$ case). We show that, when $p = 2$, for
each cluster containing an even number of roots of $f$, there are $0$, $1$, or
$2$ components of $(\mathcal{Y}^{\mathrm{rst}})_s$ corresponding to it, and we
determine a direct method of finding and describing them. We also define a
polynomial $F(T) \in K[T]$ whose roots allow us to find the components of
$(\mathcal{Y}^{\mathrm{rst}})_s$ which are not connected to even-cardinality
clusters.
