If you only have an almost complex manifold with a compatible metric i.e. an almost Hermitian manifold $(M,g,J)$ then you can define the Ricci form globally by $$\rho(X,Y):=Ric(JX,Y).$$ And vanishing of $Ric$, $\rho$ are equivalent. You don't need your manifold to be complex or Kähler.
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Correction of terminology. Hermitian manifolds have an integrable almost complex structure. Relaxing the integrability assumption gives an almost Hermitian manifold.
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