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

I would like some construction of "$\infty$-constructible sheaves" which includes both this category and also constructible complexes of 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?