Codazzi tensor fields in reductive homogeneous spaces

We extend the results about left-invariant Codazzi tensor fields on Lie groups equipped with left-invariant Riemannian metrics obtained by d’Atri in 1985 to the setting of reductive homogeneous spaces $G\mathrm{/}H$, where the curvature of the canonical connection of second kind associated with the fixed reductive decomposition $\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{m}$ enters the picture. In particular, we show that invariant Codazzi tensor fields on a naturally reductive homogeneous space are parallel.