There are no genuine Kähler-Codazzi manifolds

Nearly Kähler- and Kähler-Codazzi-type manifolds are defined in a very similar way. We prove that nearly Kähler-type manifolds make sense only in Hermitian and para-Hermitian contexts, and that Kähler-Codazzi-type manifolds reduce to Kähler-type manifolds in all the four Hermitian, para-Hermitian, Norden and product Riemannian geometries. Kähler-Codazzi condition is also studied on almost complex golden manifolds.