Given a Galois cover \(Y \rightarrow X\) 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 \(\mathcal {Y}^{\textrm{rst}}\) of Y, and and we discuss its properties, focusing on the case when \(Y: y^2 = f(x)\) is a hyperelliptic curve viewed as a degree-2 cover of the projective line \(X{:}{=} \mathbb {P}_K^1\) . Over residue characteristic different from 2, it follows from known results that the toric rank (i.e. the number of loops in the graph of components) of the special fiber of \(\mathcal {Y}^{\textrm{rst}}\) can be computed directly from the knowledge of the even-cardinality clusters of roots of the defining polynomial f. We instead consider the “wild” case of residue characteristic 2 and demonstrate an analog to this result, showing that each even-cardinality cluster of roots of f gives rise to a loop in the graph of components of the special fiber of \(\mathcal {Y}^{\textrm{rst}}\) if and only if the depth of the cluster exceeds some threshold, and we provide a computational description of and bounds for that threshold. As a bonus, our framework also allows us to provide a formula for the 2-rank of the special fiber of \(\mathcal {Y}^{\textrm{rst}}\) .
