Real Betti realization is a symmetric monoidal functor from the category of motivic spectra to that of topological spectra, extending the functor that associates to a scheme over R the space of its real points. In this article, we prove some results about the real Betti realizations of certain motivic E₁- and E_-rings. We show that the motivic Thom spectrum functor and the topological one correspond to each other, as symmetric monoidal functors, under real realization. In particular, we obtain equivalences of E_-rings between the real realizations of the variants MGL, MSL, and MSp of algebraic cobordism, and the variants MO, MSO, and MU of topological cobordism, respectively. Using this, we identify the E₁-ring structure on the real realization of ko, the very effective cover of Hermitian K-theory, by an explicit 2-local fracture square, as being equivalent to L (R) ₀, the connective L-theory spectrum of R.
Julie Bannwart (Mon,) studied this question.