Recovering a compact Hausdorff space $X$ from the compatibility ordering on $C(X)$

Let $f$ and $g$ be scalar-valued, continuous functions on some topological space. We say that $g$ dominates $f$ in the compatibility ordering if $g$ coincides with $f$ on the support of $f$. We prove that two compact Hausdorff spaces are homeomorphic if and only if there exists a compatibility isomorphism between their families of scalar-valued, continuous functions. We derive the classical theorems of Gelfand-Kolmogorov, Milgram and Kaplansky as easy corollaries to our result as well as a theorem of Jarosz [Bull. Canad. Math. Soc. 1990] thereby building~a common roof for these theorems. Sharp automatic-continuity results for compatibility isomorphisms are also established. Added on 30.03.2021: Unfortunately, Theorem 1.1 of the present manuscript is flawed. Erratum and addendum written jointly with D. H. Leung is attached. Besides providing an amendment to the said statement, it also contains more complete proof of Proposition 4.1. Theorems 1.2--1.3 are unaffected.