  We construct a relative version of topological $K$-theory of dg categories
over an arbitrary quasi-compact, quasi-separated $\mathbb{C}$-scheme $X$. This
has as input a $\text{Perf}(X)$-linear stable $\infty$-category and output a
sheaf of spectra on $X(\mathbb{C})$, the space of complex points of $X$. We
then characterize the values of this functor on inputs of the form
$Mod_{A}^{\omega}$, for $A$ a derived Azumaya algebra over $X$. In such cases
we show that this coincides with the $\alpha$-twisted topological $K$-theory of
$X(\mathbb{C})$ for some appropriately defined twist of $K$-theory. We use this
to provide a topological analogue of a classical result of Quillen's on the
algebraic $K$-theory of Severi-Brauer varieties.
