Chern connection and Levi Civita connection on Kahler manifold

I'm reading Voisin's book on complex geometry. A theorem states on Kahler manifold, Chern connection and Levi Civita connection coincide on (1,0). But I feel difficult to formalize it rigorously.

Let $X$ be a Kahler manifold. Let $TX$ be its holomorphic tangent bundle and $T_{\mathbb R}X$ be its real tangent bundle. Then, in my opinion, (1,0) part of chern connection is a map: $$ \nabla^{1,0}: \Gamma(X,TX)\rightarrow \Gamma(\Omega^{1,0}_X\otimes_{\mathbb C}TX) $$ and Levi Civita connection is a map: $$ \nabla_{LC}: \Gamma(X,T_{\mathbb R}X)\rightarrow \Gamma(X,T^*_\mathbb RX\otimes_{\mathbb R}T_\mathbb RX) $$

My question:

(1) How to identify $T_{\mathbb R}X$ with $TX$ and $\Omega_X^{1,0}$ with $T^*X$?

(2) Why $\Omega^{1,0}_X\otimes_{\mathbb C}TX = T^*_\mathbb RX\otimes_{\mathbb R}T_\mathbb RX$? It seems weird because one is $\mathbb R$-tensor but another is $\mathbb C$-tensor.

Thank you in advance!

Let $T^{1,0}X$ be the holomorphic tangent bundle, and $T_{\mathbf{C}}X=T_{\mathbf{R}}X\otimes\mathbf{C}=T^{1,0}X\oplus\overline{T^{1,0}X}$ be the complex tangent bundle.

By complexification the Levi-Civita connection extends to a complex connection $$\nabla_{LC}: \Gamma(X,T_{\mathbf{C}}X)\rightarrow \Gamma(X,T_{\mathbf{C}}^*X\otimes T_{\mathbf{C}}X).$$ The Chern connection is $$\nabla^{1,0}: \Gamma(X,T^{1,0}X)\rightarrow\Gamma(X,T_{\mathbf{C}}^*X\otimes T^{1,0}X).$$ Since $T^{1,0}X\subset T_{\mathbf{C}}X$, there are natural inclusions $\Gamma(X,T^{1,0}X)\subset \Gamma(X,T_{\mathbf{C}}X)$ and $\Gamma(X,T_{\mathbf{C}}^*X\otimes T^{1,0}X)\subset \Gamma(X,T_{\mathbf{C}}^*X\otimes T_{\mathbf{C}}X)$ which give meaning to the theorem.