We investigate the incoherent beams with two orthogonal polarizations in nonlocal nonlinear media, one of which is a fundamental Gaussian beam and the other is spiraling elliptic Hermite–Gaussian beam carrying the orbital angular momentum (OAM). Using the variational approach, we obtain the critical power and the critical OAM required for the vector spiraling elliptic Hermite–Gaussian solitons. In the strong nonlocality region, two components of the vector beam contribute to the nonlinear refractive index in a linear manner by the sum of their respective power. The nonlinear refractive index exhibits a circularly symmetrical profile in despite of the elliptic shapes for spiraling Hermite–Gaussian beams. We find that in the strong nonlocality region, the critical power and the rotational velocity are the same regardless of the relative ratio of the constituent powers. The nonlinear refractive index loses its circular symmetry in weak nonlocality region, and the nonlinear coupling effect is observed. Due to the radiation of the OAM, the damping of the rotation is predicted, and can be suppressed by decreasing the proportion of the spiraling elliptic component of the vector beam.
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