  We study metric-compatible Poisson structures in the semi-classical limit of
noncommutative emergent gravity. Space-time is realized as quantized symplectic
submanifold embedded in R^D, whose effective metric depends on the embedding as
well as on the Poisson structure. We study solutions of the equations of motion
for the Poisson structure, focusing on a natural class of solutions such that
the effective metric coincides with the embedding metric. This leads to
i-(anti-) self-dual complexified Poisson structures in four space-time
dimensions with Lorentzian signature. Solutions on manifolds with conformally
flat metric are obtained and tools are developed which allow to systematically
re-derive previous results, e.g. for the Schwarzschild metric. It turns out
that the effective gauge coupling is related to the symplectic volume density,
and may vary significantly over space-time. To avoid this problem, we consider
in a second part space-time manifolds with compactified extra dimensions and
split noncommutativity, where solutions with constant gauge coupling are
obtained for several physically relevant geometries.
