  We study the hypersymplectic geometry of the moduli space of solutions to
Hitchin's harmonic map equations on a $G$-bundle. This is the split-signature
analogue of Hitchin's Higgs bundle moduli space. Due to the lack of
definiteness, this moduli space is globally not well-behaved. However, we are
able to construct a smooth open set consisting of solutions with small Higgs
field, on which we can investigate the hypersymplectic geometry. Finally, we
reinterpret our results in terms of the Riemannian geometry of the moduli space
of $G$-connections.
