  We give a condition for a hyperelliptic curve $C$ over a local field $K$ to
be locally soluble, on the condition that $C$ obtains semistable reduction
after a tame extension of $K$, and that the residue field $k$ is sufficiently
large relative to the genus of the curve. The condition is presented in terms
of the cluster picture of $C$, a combinatorial object which determines much of
the local arithmetic of $C$.
