Wick rotations in deformation quantization

We study formal and non-formal deformation quantizations of a family manifolds that can be obtained by phase space reduction from $\mathbb{C}^{1+n}$ with the Wick star product in arbitrary signature. Two special cases such are complex projective $\mathbb{CP}^n$ hyperbolic disc $\mathbb{D}^n$. generalize several older results to this setting: The construction products their explicit description bidifferential operators, existence convergent subalgebra "polynomial" functions, its completion an algebra certain analytic functions allow easy characterization via holomorphic extensions. Moreover, we find isomorphism between for different signatures, linking e.g. on More precisely, describe (polynomial or analytic) function algebras is compatible Poisson brackets products. This essentially given rotation, i.e. extension restriction new domain. It not *-involution pointwise conjugation.