  By complexifying a Hamiltonian system one obtains dynamics on a holomorphic
symplectic manifold. To invert this construction we present a theory of real
forms which not only recovers the original system but also yields different
real Hamiltonian systems which share the same complexification. This provides a
notion of real forms for holomorphic Hamiltonian systems analogous to that of
real forms for complex Lie algebras. Our main result is that the
complexification of any analytic mechanical system on a Grassmannian admits a
real form on a compact symplectic manifold. This produces a `unitary trick' for
Hamiltonian systems which curiously requires an essential use of hyperk\"ahler
geometry. We demonstrate this result by finding compact real forms for the
simple pendulum, the spherical pendulum, and the rigid body.
