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We define united K-theory for real C*-algebras, generalizing Bousfield'stopological united K-theory. United K-theory incorporates three functors --real K-theory, complex K-theory, and self-conjugate K-theory -- and the naturaltransformations among them. The advantage of united K-theory over ordinaryK-theory lies in its homological algebraic properties, which allow us toconstruct a Kunneth-type, non-splitting, short exact sequence whose middle termis the united K-theory of the tensor product of two real C*-algebras A and Bwhich holds as long as the complexification of A is in the bootstrap category.Since united K-theory contains ordinary K-theory, our sequence provides a wayto compute the K-theory of the tensor product of two real C*-algebras.  As an application, we compute the united K-theory of the tensor product oftwo real Cuntz algebras. Unlike in the complex case, it turns out that theisomorphism class of the tensor product O_{k+1} otimes O_{l+1} is notdetermined solely by the greatest common divisor of k and l. Hence we haveexamples of non-isomorphic, simple, purely infinite, real C*-algebras whosecomplexifications are isomorphic.

Real C*-Algebras, United K-Theory, and the Künneth Formula