$q$-Hodge complexes and refined $\operatorname{TC}^-$

As a consequence of Efimov's proof of rigidity of the $\infty$-category of localising motives, Efimov and Scholze have constructed refinements of localising invariants such as $\operatorname{THH}$ and $\operatorname{TC}^-$. These refinements often contain vastly more information than the original invariant. In this article we explain a general recipe how to compute the refinements in certain situations. We then apply this recipe to compute the homotopy groups of $\operatorname{TC}^{-,\mathrm{ref}}(\mathrm{ku}\otimes\mathbb Q/\mathrm{ku})$ and $\operatorname{TC}^{-,\mathrm{ref}}(\mathrm{KU}\otimes\mathbb Q/\mathrm{KU})$. The result has a rather surprising geometric description and contains non-trivial information modulo any prime, in contrast to the unrefined $\operatorname{TC}^-$.