  We elaborate on nonassociative differential geometry of phase spaces endowed
with nonholonomic (non-integrable) distributions and frames, nonlinear and
linear connections, symmetric and nonsymmetric metrics, and correspondingly
adapted quasi-Hopf algebra structures. The approach is based on the concept of
nonassociative star product introduced for describing closed strings moving in
a constant R-flux background. Generalized Moyal-Weyl deformations are
considered when, for nonassociative and noncommutative terms of star
deformations, there are used nonholonomic frames (bases) instead of local
partial derivatives. In such modified nonassociative and nonholonomic
spacetimes and associated complex/ real phase spaces, the coefficients of
geometric and physical objects depend both on base spacetime coordinates and
conventional (co) fiber velocity/ momentum variables like in (non) commutative
Finsler-Lagrange-Hamilton geometry. For nonassociative and (non) commutative
phase spaces modelled as total spaces of (co) tangent bundles on Lorentz
manifolds enabled with star products and nonholonomic frames, we consider
associated nonlinear connection, N-connection, structures determining
conventional horizontal and (co) vertical (for instance, 4+4) splitting of
dimensions and N-adapted decompositions of fundamental geometric objects. There
are defined and computed in abstract geometric and N-adapted coefficient forms
the torsion, curvature and Ricci tensors. We extend certain methods of
nonholonomic geometry in order to construct R-flux deformations of vacuum
Einstein equations for the case of N-adapted linear connections and symmetric
and nonsymmetric metric structures.
