Let \(A\) and \(B\) be commutative Banach algebras. A map \(T: A \to B\) is called peripherally multiplicative if \(\sigma_{\pi}(Tf \cdot Tg)= \sigma_{\pi}(f\cdot g)\) for all \(f, g\in A\), where \(\sigma_{\pi}(f)\) is the peripheral spectrum of \(f\) defined by \(\sigma_{\pi}(f)= \{\lambda \in \sigma(f): |\lambda|= \max_{z\in \sigma(f)} |z|\}\) (here, \(\sigma(f)\) is the spectrum of \(f\)). In the case where \(A\) and \(B\) are natural Banach function algebras on compact Hausdorff spaces, the author shows that every peripherally multiplicative operator \(T: A \to B\) is injective. As a main result of the paper, he shows that, if \(T\) is unital (i.e., \(T1= 1\)) and surjective, then there is a homeomorphism \(\tau\) from \(\partial (A)\), the Šilov boundary of \(A\), to \(\partial(B)\) such that \(|f(x)|= |Tf(\tau(x))|\) for all \(f\in A\) and \(x\in \partial(A)\).