  Let $(M,g,J)$ be a Riemannian manifold with a compatible integrable complex
structure $J\in\mathrm{End}(T_\mathbb{R} M)$ and $\mathcal{A}_{g,J}$ be the
space of real connections on $T_\mathbb{R} M$ preserving both $g$ and $J$. In
this paper, we investigate the relationship between the geometry of real
connections in $\mathcal{A}_{g,J}$ and that of Hermitian connections on
$T^{1,0}M$. In particular, we study the geometry of the real Chern connection
$\nabla^{\mathrm{Ch,\mathrm{R}}}$ on $(M,g,J)$, and obtain K\"ahler-Einstein
metrics by using real Chern-Einstein metrics.
