I don't know if this is an answer, but at least one can build examples with all leaves closed.

If one picks C such that its Jacobian is a product of several complex tori, one of which is an elliptic curve. For instance one could take C of genus 2 such that its jacobian is a product $E_1 \times E_2$ with the $E_i$ elliptic curves.

Then one considers the map $F : E_1 x C \to E_1$ obtained by composing the natural maps

$E_1 \times C \to E_1 \times E_1 \times E_2 \to E_1 \times E_1 \to E_1$ where the last arrow is the sum.

The pull-back by F of a nonzero holomorphic form on $E_1$ should give an example.

I don't know if this is an answer, but at least one can build examples with all leaves closed.

If one picks C such that its Jacobian is a product of several complex tori, one of which is an elliptic curve. For instance one could take C of genus 2 such that its jacobian is a product $E_1 \times E_2$ with the $E_i$ elliptic curves.

Then one considers the map $F : E_1 \times C \to E_1$ obtained by composing the natural maps

$E_1 \times C \to E_1 \times E_1 \times E_2 \to E_1 \times E_1 \to E_1$ where the last arrow is the sum.

The pull-back by F of a nonzero holomorphic form on $E_1$ should give an example.