I'm aware that mathematically speaking, Calabi-Yau manifolds are complex manifolds with vanishing first Chern number. However from physics point of view, Calabi-Yau manifolds are related to the solution of Einstein's field equation in vacuum environment (i.e with vanishing stress–energy tensor). Since Einstein's field equation is on a 4-dimensional real manifold, why Calabi-Yau manifolds are complex? Is there a "real version" of Calabi-Yau manifold?