As Umat says, If a local orientation is used at the same point as user subroutine UMAT, the stress and strain components will be in the local orientation; and, in the case of finite-strain analysis, the basis system in which stress and strain components are stored rotates with the material.

The rotation matrix is obtained by polar decomposition F=RU=VR, and I know the deformaton gradient passes into UMAT  F_umat=R'*F*R, where F is messured in the global coordinate(assuming initial local coordinate coincide with the global coordinate). But there is an error with the DG output by abaqus, Have you noticed this issue?