Reconstructing complexes of sheaves from their cohomology sheaves

If $R$ is an algebra over some field $k$, and $C$ is a complex of modules over $R$, then according to B. Keller's Introduction to A-infinity algebras and modules'', one can record the isomorphism class of $C$ in the derived category $D(R)$ by equipping the cohomology $H(C)$ with the structure of an $A$-infinity $R$-module. This means you give $k$-linear (degree $1-n$) maps $H(C) \otimes R^{\otimes n-1} \to H(C)$, satisfying some incomprehensible condition.

In particular, if $C$ has finite dimensional cohomology groups of bounded degrees, and $A$ is finite dimensional, the amount of information required to specify $C$ is a priori bounded.

I want to know if the same can be done for sheaves, i.e.:

If $X$ is some reasonable space, and $C$ is a complex of sheaves on $X$, can I recover the isomorphism class of $C$ inside the derived category $D(X)$ just from the cohomology sheaves $H^i(C)$, plus some additional structure?

In fact I am mostly interested in the case where the sheaves are constructible with respect to a fixed stratification. The point is that I have some space $X$ with a Whitney stratification $S$; it is not so mysterious what the possible cohomology sheaves of an element of $D_S(X)$ could be, and I want to be able to say, in terms of those, here is an exhaustive list of all the possible isomorphism classes of elements in $D_S(X)$.

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