  Joyce structures were introduced by T. Bridgeland in the context of the space
of stability conditions of a three-dimensional Calabi-Yau category and its
associated Donaldson-Thomas invariants. In subsequent work, T. Bridgeland and
I. Strachan showed that Joyce structures satisfying a certain non-degeneracy
condition encode a complex hyperk\"{a}hler structure on the tangent bundle of
the base of the Joyce structure. In this work we give a definition of an
analogous structure over an affine special K\"{a}hler (ASK) manifold, which we
call a special Joyce structure. Furthermore, we show that it encodes a real
hyperk\"{a}hler (HK) structure on the tangent bundle of the ASK manifold,
possibly of indefinite signature. Particular examples include the semi-flat HK
metric associated to an ASK manifold (also known as the rigid c-map metric) and
the HK metrics associated to certain uncoupled variations of BPS structures
over the ASK manifold. Finally, we relate the HK metrics coming from special
Joyce structures to HK metrics on the total space of algebraic integrable
systems.
