Several physicists consider non-Poisson bivectors but still apply Kontsevich formality in order to get deformation quantization type results: see e.g. Szabos's review arxiv.org/pdf/1903.05673.pdf

The result they get is a `non-associative star product’.

What is the algebraic structure that the get, or, how to state correctly the deformation quantization problem in this non-associative and non-Poisson context?