noncommutative deformations of complex manifolds) [@MR1855264; @MR1384750; @KS1; @KS2; @MR2348030; @Tsygan1; @Y1]. The quantization of the sheaf of holomorphic functions ${\mathcal{O}_{{\boldsymbol{X}}}}$ of a complex manifold ${\boldsymbol{X}}$ may no longer produce a sheaf of algebras but, instead, lead to a nonabelian gerbe over the complex manifold ${\boldsymbol{X}}$ [@MR2348030; @Y2] or, in Kontsevich’s terminology, a stack of algebroids. Roughly speaking, an algebroid *à la* Kontsevich consists of an open cover $\{U_i\}_{i\in I}$ of the complex manifold ${\boldsymbol{X}}$, a sheaf of associative unital algebras ${\mathscr{A}}_i$ on each $U_i$, an isomorphism of algebras $g_{ij}:{\mathscr{A}}_j|_{U_{ij}}\to{\mathscr{A}}_i|_{U_{ij}}$ for each nonempty intersection $U_{ij}$, and an invertible element $a_{ijk}\in \Gamma (U_{ijk}, {\mathscr{A}}_i^{\times} )$ for each triple intersection $U_{ijk}$. The isomorphisms $g_{ij}$ do not satisfy the usual cocycle condition. Instead, the equations $g_{ij}\circ g_{jk}\circ g_{ki}=\operatorname{Ad}_{a^{-1}_{ijk}}$ are satisfied as well as other compatibility conditions (among which a “tetrahedron equation”). In the terminology of [@LSX:adv], an algebroid *à la* Kontsevich would be described as an extension of a [Čech ]{}groupoid by algebras. A stack of algebroids can be thought of as a Morita equivalence class (see [@LSX:adv]) of algebroids. A canonical abelian category of coherent sheaves can be defined on a quantum complex manifold using its stack of algebroids description [@MR1855264; @MR1384750; 