Several authors used quasi-Hopf algebras in the treatment of symmetries
for nonassociative deformations. In associative case, Drinfeld
twists in the standard sense of Hopf algebras can explain only very special
star products, while other twists have to be treated in the wider sense of
(associative) bialgebroids (Drinfeld-Xu twists). Analogously, quasi-Hopf
algebras are a rather restrictive framework in the nonassociative case
and it is an open question what should be the notion of a quasi-bialgebroid (or
even quasi-Hopf algebroid) allowing for general nonassociative twists.
After reviewing usage of Hopf algebroids over a noncommutative base algebra
for the description of noncommutative
phase spaces and associative star products,
I will present my attempt to simultaneously and self-consistently
twist the monoidal category of bimodules over the
base algebra and twist the bialgebroid to obtain a generalized version of
a bialgebroid which is an internal bialgebroid
in the twisted monoidal category of bimodules. In other words,
it is still associative in the twisted category. The twisted version
can be framed within an axiomatics without referring to twists.