  Using the framework for multiplicative parametrized homotopy theory
introduced in joint work with C. Schlichtkrull, we produce a multiplicative
comparison between the homotopical and operator algebraic constructions of
twisted K-theory, both in the real and complex case. We also improve several
comparison results about twisted K-theory of C*-algebras to include
multiplicative structures. Our results can also be interpreted in the
infinity-categorical setup for parametrized spectra.
