Apparently, an example of a pivotal non-spherical category can be found in the world of Hopf algebras/representation theory:
- Let $(H, \omega)$ be a pivotal non-spherical Hopf algebra with pivot $\omega$ (see From Hopf algebras to tensor categories by Andruskiewitsch et al. for an example). Then the action of the pivot $\omega$ endows the category of finite-dimensional $H$-modules with a pivotal non-spherical structure.
What other examples of pivotal non-spherical categories are known? In particular, I am interested in examples where the category is even a fusion category.