  We give a comparative description of the Poisson structures on the moduli
spaces of flat connections on real surfaces and holomorphic Poisson structures
on the moduli spaces of holomorphic bundles on complex surfaces. The symplectic
leaves of the latter are classified by restrictions of the bundles to certain
divisors. This can be regarded as fixing a "complex analogue of the holonomy"
of a connection along a "complex analogue of the boundary" in analogy with the
real case.
