Let $(X,d)$ be a compact metric space, $\tau $ be a topological involution on $(X, d)$ and $\alpha \in (0, 1]$. We prove that every weakly compact composition operator on real Lipschitz spaces of complex-valued functions ${\rm Lip}(X, d^\alpha, \tau)$ and ${\rm lip}(X, d^\alpha, \tau)$ is compact.
