In this paper, we study order isomorphisms between $C(X,\tau)$ and $C(Y,\eta)$ where $X$ and $Y$ are compact Hausdorff spaces and $\tau$ and $\eta$ are topological involutions on $X$ and $Y$, respectively. We first give some sufficient conditions that a map $T$ from $C(X,\tau)$ to $C(Y,\eta)$ be an order isomorphism. We next characterize the structure of order isomorphisms $T$ with certain properties between these algebras. Finally, we prove that every order isomorphism $T$ from $C_\mathbb{R}(X)$ to $C_\mathbb{R}(Y)$ is an essential weighted composition operator.d
