I'm reading the book "Tensor categories" by Etingoff (and others). In remark 2.8.6 (posted below), it is claimed that the category $\mathcal{C}_G^\omega$ (defined in example 2.3.8, also below) is not isomorphic to a strict tensor category whenever $\omega$ is not cohomologically trivial. Can someone explain this a bit more?
If someone knows another example of a tensor category that is not isomorphic to a strict tensor category, I'm also interested in that.
Edit: When no answer is given here after the bounty period ends, I will post this question on MO.