I think I understand the last paragraph (although I believe you meant $\pi_{*}L$ and $\pi_{*} \mathcal{O}$, if I'm not mistaken) but I'm not seeing how your reformulation of the statement is equivalent. Can you explain this briefly?

I see; I was over-complicating this. So the idea is that $H^{0}(C, \Omega_{C})^{*}$ decomposes as an $\iota$-representation. The subspace where the action is trivial corresponds to forms pulled back from the quotient, and the subspace where $\iota$ acts non-trivially is the tangent space of the Prym. This is what you're getting at, correct? So this implies the global action on the Prym is by $\pm 1$.

Thanks a lot for the reply. My surfaces are projective, so it is not totally obvious to me where a manifold with boundary might enter the picture. But some of these papers on shadows of a lattice look intriguing and relevant somehow!

I would be very interested in both!