We construct an ungraded version of Beilinson–Ginzburg–Soergel's Koszul duality for Langlands dual flag varieties, inspired by Beilinson's construction of rational motivic cohomology in terms of K $K$ -theory. For this, we introduce and study categories DK S ( X ) $\hbox{DK}_\mathcal {S}(X)$ of S $\mathcal {S}$ -constructible K $K$ -motivic sheaves on varieties X $X$ with an affine stratification. We show that there is a natural and geometric functor, called Beilinson realisation, from S $\mathcal {S}$ -constructible mixed sheaves D S m i x ( X ) $\hbox{D}^{mix}_\mathcal {S}(X)$ to DK S ( X ) $\hbox{DK}_\mathcal {S}(X)$ . We then show that Koszul duality intertwines the Betti realisation and Beilinson realisation functors and descends to an equivalence of constructible sheaves and constructible K $K$ -motivic sheaves on Langlands dual flag varieties.