We first show that a bounded linear operator $T$ on a real Banach space $E$ is weakly compact if and only if the complex linear operator $T'$ on the complex Banach space $E_{\mathbb{C}}$ is weakly compact, where $E_{\mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{\mathbb{C}}$ associated with $T$. Next we show that every weakly compact composition operator on real Lipschitz spaces of complex-valued functions on compact metric spaces with Lipschitz involutions is compact.X
