Let $X$ be a space and $\mathcal U$ a cover of $X$. Instead of Čech cohomology, I would like to take the following construction:
let 
$$I= \{ \text{finite nonempty intersections of elements of }\,\mathcal U\},$$
which is a poset, and now I can take the nerve $N(I)$ in the sense of category theory,
whose $n$-simplices are chains of length $n+1$ in $N(I)$.
Given a sheaf $\mathcal F$ on $X$, I get a local system of coefficients $F$ on $N(I)$ by taking $F(\sigma) = \mathcal F(\min \sigma)$.
I would like to relate $H^*(N(I), F)$ to the sheaf cohomology $H^*(X,\mathcal F)$. Do you know how to do this? 

My hope is that this is sheaf cohomology if we assume enough acyclicity about $\mathcal F$ and $I$. This complex smells similar to the Čech complex, but I am not sure of the general relation. If $X$ was a simplicial complex and $\mathcal U$ the covering by star neighborhoods, then we recover Čech cohomology with respect to star neighborhoods in the barycentric subdivision.

In my situation $X$ is a quasi-compact separated scheme, $\mathcal U$ is a cover by affine opens, and $\mathcal F$ is a quasi-coherent sheaf. But, the same setup might work if we just assume $\mathcal F$ is acyclic on every element of $I$.