  We discuss the appearance of relativistic geometric quantum phases for a
Dirac neutral particle based on possible scenarios of the Lorentz symmetry
violation background in the CPT-even gauge sector of Standard Model Extension.
Relativistic analogues of the Anandan quantum phase [J. Anandan, Phys. Lett. A
138, 347 (1989)] are obtained based on the the parity-even and parity-odd
sectors of the tensor $\left(K_{F}\right)_{\mu\nu\alpha\beta}$.
