Several physicists consider non-Poisson bivectors but still apply [Kontsevich formality][1] in order to get deformation quantization type results: see e.g. Szabos's review [An introduction to nonassociative physics][2].
 
The result they get is a `non-associative star product’. 
 
What is the algebraic structure that they get, or, how to state correctly the deformation quantization problem in this non-associative and non-Poisson context?


  [1]: https://ncatlab.org/nlab/show/Kontsevich+formality
  [2]: https://arxiv.org/abs/1903.05673