  We study tautological classes on the moduli space of stable $n$-pointed
hyperelliptic curves of genus $g$ with rational tails. Our result gives a
complete description of tautological relations. The method is based on the
approach of Yin in comparing tautological classes on the moduli of curves and
the universal Jacobian. It is proven that all relations come from the Jacobian
side. The intersection pairings are shown to be perfect in all degrees. We show
that the tautological algebra coincides with its image in cohomology via the
cycle class map. The latter is identified with monodromy invariant classes in
cohomology. The connection with recent conjectures by Pixton is also discussed.
