In this paper, we study surjective $\mathbb{R}^+$-homogeneous norm-additive in modulus between $ C(X, \tau)$-algebras. We first show that if $ X $ and $ Y $ are compact Hausdorff spaces, $ \tau $ and $ \eta $ are topological involutions on $ X $ and $ Y $, respectively, and $ T: C(X,\tau) \longrightarrow C(Y, \eta) $ is a surjective $\mathbb{R}^+$-homogeneous norm-additive in modulus, then there exists a unique bijective map $ \Phi : Y_{\eta} \longrightarrow X_{\tau} $ such that $ |T(f)(y)| = |f(x)| $ for all $ f \in C(X, \tau)$, $y \in Y $ and $ x \in \Phi(y_{\eta}) $, 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 \} $. We next show that if $ T: C_{\mathbb{R}}(X) \longrightarrow C_{\mathbb{R}}(Y)$ is a surjective $\mathbb{R}^+$-homogeneous norm-additive in modulus, then there exists a homeomorphism $ \varphi : Y \longrightarrow X $ such that $ |T(f)(y)| = |f(\varphi(y))| $ for all $ f \in C(X,\tau) $ and $ y \in Y $, where $ X $ and $ Y $ are compact Hausdorff spaces.d
