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In this work, we present a variational and quantitative phase-field model for non-isothermal sintering processes. The model is derived via an extended non-diagonal phase-field model. The model evolution equations have naturally cross-coupling terms between the conserved kinetics (i.e., mass and thermal transfer) and the non-conserved one (grain growth). These terms are shown via asymptotic analysis to be instrumental in ensuring the elimination of interface artefacts, while also examined to not modify the thermodynamic equilibrium condition (characterized by dihedral angle). Moreover, we demonstrate that the trapping effects and existence of surface diffusion in conservation laws are direction-dependent. An anisotropic interpolation scheme of the kinetic mobilities which differentiates the normal and the tangential directions along the interface is discussed. Numerically, we demonstrate the importance of the cross-couplings and the anisotropic interpolation via presenting thermal-microstructural evolutions.