Let A be a real B*-algebra containing a p*-subalgebra that is isomorphic to the real quaternion algebra H. Suppose the spectrum of every self-adjoint element in A is contained in the real line. Then it is proved that there exists a quaternionic Hubert space X and an isometric isomorphism n of A onto a closed *-subalgebra of BL(X), the algebra of all bounded linear operators on X. If, in addition to the above hypotheses, every element in A is normal, then A is also proved to be isometrically isomorphic to C(Y, E), the algebra of all continuous H-valued functions on a compact HausdorfF space Y. © 1994 American Mathematical Society.