  The moduli space of Higgs bundles can be constructed as a quotient of an
infinite-dimensional space and hence admits an orbit type decomposition. In
this paper, we show that the orbit type decomposition is a complex Whitney
stratification such that each stratum is a complex symplectic submanifold and
hence admits a complex Poisson bracket. Moreover, these Poisson brackets glue
to a Poisson bracket on the structure sheaf of the moduli space so that the
moduli space is a stratified complex symplectic space
