For a compact Hausdorff space X, let C(X, H) denote the set of all quaternion-valued functions on X. It is proved that if a real B* -algebra A satisfies the following conditions: (i) the spectrum of every selfadjoint element is contained in the real line and (ii) every element in A is normal, then A is isometrically *-isomorphic to a closed *-subalgebra of C(X, H) for some compact Hausdorff X. In particular, a real C*-algebra in which every element is normal is isometrically *-isomorphic to a closed *-subalgebra of C(X, H). © 1992 American Mathematical Society.