Suppose $X$ is a topological space and $k$ some discrete coefficient field. Let's define an $\infty$-local system on $X$ to be a representation of the DG ring $C_*(\Omega X,k)$ of chains of loops on $X$ into $DG$-vect. (I think this is equivalent, at least when $\text{char}(k)=0$, to a locally constant sheaf where "sheaf" is defined in an appropriate higher-topos sense).

I would like some construction of "$\infty$-constructible sheaf" which includes both this category and also complexes of constructible sheaves for some fixed stratification of $X$. Do topologists know of one? Is there an analogue of the Riemann-Hilbert correspondence for $X$ a complex algebraic manifold?