Clearly looking at sheaves on the geometric realisation gives something too far
removed from the simplicial picture. This is essentially because there are too
many sheaves on a simplex have (most of which are unrelated to simplicial
ideas). What one could do is to consider such sheaves which are constructible
with respect to the skeleton filtration, i.e., are constant on each open
simplex. This can be described inductively using Artin gluing. I think it
amounts to the following for a simplicial set $F$.

For each simplex $c\in F_n$ we have a set $T_c$, the constant value of the sheaf
$T$ on the interior of the simplex corresponding to $c$.

For each surjective map $f\colon [n] \to [m]$ in $\Delta$ the corresponding (degeneracy) map
on geometric simplices maps the interior of $\Delta_n$ into (onto in fact) the interior of $\Delta_m$ and
hence we have a bijection $T_{f(c)} \to T_c$. These bijections are transitive with
respect to compositions of $f$'s.

For each injective map $f\colon [m] \to [n]$ in $\Delta$ the corresponding map on geometric
simplices maps $\Delta_m$ onto a closed subset of $\Delta_n$. If $j\colon \Delta^o_n \hookrightarrow \Delta_n$ is the
inclusion of the interior we get an adjunction map $T \to j_\ast j^\ast T$ and
$j_\ast j^\ast T=T_c$ where $T_c$ also denotes the constanct sheaf with value $T_c$. If
$f'\colon \Delta^o_m \hookrightarrow \Delta_n$ is the inclusion of the interior composed with $f$ we can
restrict the adjunction map to get a map $T_{f(c)}=f'^\ast T \to f'T_c$ and taking
global sections we get an actual map $T_{f(c)} \to T_c$. These maps are transitive
with respect compositions of $f$'s.

We have a compatibility between maps coming from surjections and
injections. Unless something very funny is going on this compatibility should be
that we wind up with a function on the comma category $\Delta/F$ which takes
surjections $[n] \to [m]$ to isomorphisms.

There is the stronger condition on the sheaf $F$, namely that it is constant on
each star of each simplex. This means on the one hand that it is locally
constant on the geometric realisation, on the other hand that $T_{f(c)} \to T_c$
is always an isomorpism.