  Let k be a field not of characteristic two and L be a set of almost all
rational primes invertible in k. Suppose we have a variety X/k and strictly
compatible system {M_ell -> X : ell in L} of constructible F_ell-sheaves. If
the system is orthogonally or symplectically self-dual, then the geometric
monodromy group of M_ell is a subgroup of a corresponding isometry group G_ell
over F_ell, and we say it has big monodromy if it contains the derived subgroup
DG_ell=[G_ell,G_ell]. We prove a theorem which gives sufficient conditions for
M_ell to have big monodromy. We apply the theorem to explicit systems arising
from the middle cohomology of families of hyperelliptic curves and elliptic
surfaces to show that the monodromy is uniformly big as we vary ell and the
system.
