It is known that for a sheaf $\mathcal{F}$ on $X$, we can associate $X_\mathcal{F}$, the étalé space of $\mathcal{F}$ over $X$ such that section of $X_\mathcal{F}$ coincides with section of $\mathcal{F}$.
Is it possible to generalize this concept to derived category of sheaves over $X$ or perverse sheaves over $X$?
There are similar questions asked before, see links below. Yet they focus on abstract nonsense argument. I would like to see concrete constructions specifically for derived category or perverse sheaves.
Etalé space construction for presheaves on a Grothendieck site
How to construct the espace étalé (space of sections) for an arbitrary category?