Let $(X,d)$ and $(Y,\rho)$ be compact metric spaces, $\tau$ and $\eta$ be Lipschitz involutions on $(X,d)$ and $(Y,\rho)$, respectively, and $\alpha,\beta\in(0,1)$. We first give some sufficient conditions that a map $T$ between little real Lipschitz algebras with Lipschitz involution $\lip(X,d^\alpha,\tau)$ and $\lip(Y,\rho^\beta,\eta)$ be an order isomorphism. We next prove that if $T:\lip(X,d^\alpha,\tau)\rightarrow \lip(Y,\rho^\beta,\eta)$ is an order isomorphism with $|T(i{\rm{Im}}\,f)|\leq T(|{\rm{Im}}\,f|)$ for all $f\in \lip(X,d^\alpha,\tau)$ and $|T^{-1}(i{\rm{Im}}\,h)|\leq T^{-1}(|{\rm{Im}}\,h|)$ for all $h\in \lip(Y,\rho^\beta,\eta)$, then $T$ is a homeomorphism from $\lip(X,d^\alpha,\tau)$ with Lipschitz sum norm topology to $\lip(Y,\rho^\beta,\eta)$ with Lipschitz sum norm topology, $T(1_X)$ and $T^{-1}(1_Y)$ are nonvanishing positive functions on $Y$ and $X$, respectively, and there exists a bijective map $\Phi:Y_\eta\rightarrow X_\tau$ such that $T(f)(y)=T(1_X)(y)f(x)$ for all $y\in Y$, $x\in\Phi(y_\eta)$ and $f\in \lip(X,d^\alpha,\tau)$ with $f(x)\in\mathbb{R}$, where $x_\tau=\{x,\tau(x)\}$ for all $x\in X$, $X_\tau=\{x_\tau: x\in X\}$, $y_\eta=\{y,\eta(y)\}$ for all $y\in Y$ and $Y_\eta=\{y_\eta: y\in Y\}$. Finally, we prove that every order isomorphism between little real Lipschitz algebras $\lip_{\mathbb{R}}(X,d^\alpha)$ and $\lip_{\mathbb{R}}(Y,\rho^\beta)$ is an essential weighted composition operator.d
