Take a spin Chern-Simons TQFT, such as $U(N)$ or $SO(N)$ with odd level. Such system depends on the spin structure of the underlying manifold.
But how exactly does the theory depend on the spin structure? Say we compute the partition function (or some correlator of Wilson loops) using two different spin structures. How do these two objects differ? I would expect a rather mild dependence, such as an overall $\pm1$ sign (or some other phase). Is this correct?