The real Betti realization of very effective Hermitian K-theory

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 $\mathbb{R}$ the space of its real points. In this article, we identify the ${\mathcal{E}}_1$-ring structure on the real realization of $\mathsf{k}\mathsf{o}$, the very effective cover of Hermitian K-theory $\mathsf{K}\mathsf{O}$, by an explicit 2-local fracture square, as being equivalent to $\mathsf{L}(\mathbb{R}{)}_{\ge 0}$, the connective L-theory spectrum of $\mathbb{R}$. Along the way, we also compute the real realization of other motivic ring spectra: motivic cohomology $\mathsf{H}\mathbb{Z}$ and $\mathsf{H}\mathbb{Z}\mathrm{/}2$, the variant $\widetilde{\mathsf{H}\mathbb{Z}}$, $\mathsf{K}\mathsf{O}$, the very effective cover $\mathsf{k}\mathsf{g}\mathsf{l}$ of algebraic K-theory $\mathsf{K}\mathsf{G}\mathsf{L}$.