A sage says:

What he calls “big” others call “Lie irreducible”. Simplest examples are when the monodromy is finite, e.g.,
a family of fibre dimension zero. Higher dimensional fibres: take a one-parameter family of, say, curves,$ f:X \rightarrow C$ over
a curve $C$ as base, maybe this family has full $Sp(2g)$ as its monodromy. Now suppose $C$ has an involution, $\sigma,$ such that
the family pulled back by $\sigma$ has “nothing to do” with the original family: for example, suppose the input family has nontrivial
unipotent local monodromy at some point $P$ of $C,$ but that the pulled back family has trivial monodromy at this point $P.$ Then
the direct sum of the two local systems will have group $Sp(2g)\times Sp(2g).$ OK, now let $D$ be the quotient of $C$ by the involution,
and consider the family over D we get as the composition of $f:X \rightarrow C$ followed by the projection of $C$ to $D.$ [In terms of local
systems, we have one on $C,$ then form the induced representation.] When we pull THIS family back to $C,$ we get the sum of the
original local system and its $\sigma$ pullback, but the monodromy of $C/D$ switches the two factors.

In other words, the family
over D has monodromy group the semidirect product of $Sp(2g)\times Sp(2g)$ with the $Z/2Z$ which switches the factors, an
irred. subgroup of $Sp(4g)$, but its identity component is the reducible “diagonal” subgroup $Sp(2g)\times Sp(2g) of Sp(4g).$
Concrete example: family $y^2=x(x-1)(x-\lambda)$ over the $\lambda$ line (legendre family), involution $\lambda \mapsto -\lambda.$
Pullback family is $y^2=x(x-1)(x+\lambda).$ Input family had unip. mono. at $0$ and $1,$ pullback has unip. mono. at $0$ and $-1.$