The real and unitary topological $K$-theories are cohomology theories defined by the $\Omega$-spectra $KO$ and $K$ respectively. These are multiplicative theories with products deriving from the infinite loop structures on the spaces that comprise the ring spectra. On the level of vector bundles these products derive from the tensor product $(-)\otimes_{\mathbb{K}}(-)$ over the relevant field. Given a space $X$ and classes $\xi=[f]\in K^p(X)$, $\eta=[g] \in \widetilde{K}^q(X)$, we can either regard $\widetilde{K}^*(X)$ as the homotopy set $\left[X,\Omega^{-*} BU\right]$ and form the product $\xi\cdot\eta$ as the composition $X\wedge X\xrightarrow{f\wedge g}\Omega^{-p}BU\wedge\Omega^{-q}BU\xrightarrow{\theta_{p,q}}\Omega^{-(p+q)}BU$ for suitable maps $\theta_{p,q}$ or, alternatively, we can regard $\tilde{K}^*(X)$ as the Grothiendieck group of vector bundles over $\Sigma^{-*}X$ and form the product $\xi\cdot\eta$ as the class represented by the vector bundle $\xi\otimes\eta$.
Does symplectic $KSp$-theory have products? And how does one make sense of them?
Bott periodicity gives $\Omega^4BSp\simeq \mathbb{Z}\times BO$, so $KSp$ is a ring spectrum and up to canonical isomorphism the groups $KSp$ and $KO$ are the same. Products exist in the groups. But how does one make sense of them as there is no quaternionic tensor product? Do you treat the quaternionic bundles as real bundles, take the tensor product and then obtain the (unrelated) class of the real tensor product in the symplectic group?
