Is possible to define a warped product manifold $M=(N,g_N) \times f(F, g_F)$ where $(N, g_N)$ is a Riemannian manifold with Riemannian metric (i.e., real manifold with real structure) and $(F, g_F)$ is a complex manifold with Hermitian metric?

Is possible to consider this warped product manifold as Hermitian manifold?

Its metric should be Hermitian because $g=g_N + f^2(g_F)=Re_N + f^2(Re_F) + f^2(Im_F)$, where with $Re_N$ and $Re_F$ I mean the real part of the $N$ and $F$ metrics, respectively, and with $Im_F$ the imaginary part of $F$-metric, being complex.