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Molecular dynamics simulation is carried out to study domain structures and rheological properties of a two-dimensional phase-separating binary fluid mixture under shear flow. In the early stage of the phase separation, anisotropic composition fluctuations appear immediately after the quench. As the domain grows, the anisotropy in the composition fluctuations increases. The quenched system eventually reaches a dynamical steady state, in which anisotropic domain structures are preserved. In the steady state, the shortest characteristic length scale R⊥ of domains decreases with increasing shear rate γ̇ as R⊥∼γ̇-1/3. Stringlike domain structures are observed in the strong shear regime, whereas randomly fluctuating patterns are observed in the weak shear regime. Moreover, the excess viscosity Δη is found to decrease with increasing shear rate as Δη∼γ̇-1/2, indicating that the phase-separating fluid mixtures are highly non-Newtonian because of domain deformations.