What is the product in KO-theory in terms of concrete homogeneous spaces?

Question

It is well-known that the spaces $KO_i$ of the $\Omega$-spectrum representing KO-theory a.k.a. real K-theory are $KO_0,KO_{-1}, \ldots, KO_{-7} = BO \times \mathbb{Z}, O, O/U, U/Sp, BSp \times \mathbb{Z}, Sp, Sp/U, U/O$, repeating every eight. These are homogeneous spaces of the stable versions of classical Lie groups, where the latter refers to colimits of finite-dimensional Lie groups over the inclusion of each group into the next by extending as the identity, e.g., $U = \varinjlim_{n \to \infty} U(n)$. It is more-or-less clear how the product on topological KO-theory in degree zero comes from a map $BO \times \mathbb{Z} \times BO \times \mathbb{Z} \to BO \times \mathbb{Z}$: think of these spaces as colimits of Grassmannians and construct a map induced by tensor product of subspaces. But my question is what concrete maps do the products in other degrees correspond to? For example, the space $O/U$ can be thought of as the space of stable orthogonal almost-complex structures on $\mathbb{R}^\infty$: $O$ acts by conjugation with stabilizer $U$. The product $KO_{-2} \times KO_{-2} \to KO_{-4}$ should correspond to a map $O/U \times O/U \to BSp \times \mathbb{Z}$. How does one take two almost-complex structures and "tensor" them to obtain an element of $BSp \times \mathbb{Z}$ i.e. a half-infinte-dimesional quaternionic subspace of $\mathbb{H}^\infty$ (or whatever your preferred model for $BSp \times \mathbb{Z}$ is)? In general I'm having a very hard time finding any concrete information on the multiplication in $KO$-theory at the point-set level. I know about the model of Atiyah-Singer which uses spaces of Clifford-linear Fredholm operators, but I'm specifically interested in models where the spaces are colimits of finite-dimensional spaces. Any information at all would be appreciated.