Let $ (X,d) $ be a pointed compact metric space and $ \tau $ be a base-point preserving Lipschitz involution on $ (X,d)$. We prove that every weakly compact composition operator on real Banach spaces of complex-valued Lipschitz functions $ {\rm Lip}_0 (X, d, \tau) $ and $ {\rm lip}_0 (X, d, \tau) $ is compact.
