Let $E$ be a globally generated vector bundle on a surface $S$ of rank $r\geq 2$. By standard facts about degeneracy loci, for a general $V\in G(r,H^0(E))$ one has:

(*)the evaluation map $ev: V\otimes \mathcal{O}_S\to E$ is injective and the cokernel is a line bundle supported on a smooth curve.

Now, let $E_1$ be a subvector bundle of $E$ and assume $E_1$ is globally generated as well. Is it possible to find $V\in G(r,H^0(E))$ which satisfies (*) and such that $V\cap H^0(E_1)$ has dimension equal to the rank of $E_1$?