We give a description of 2-local real isometries between C(X, τ) and C(Y, η) where X and Y are compact Hausdorff spaces, X is also first countable and τ and η are topological involutions on X and Y , respectively. In particular, we show that every 2-local real isometry T from C(X, τ) to C(Y, η) is a surjective real linear isometry whenever X is also separable.d
