In order to investigate the propagation characteristics of electromagnetic elliptical multi-Gaussian-Schell mode (EEMGSM) beams, based on the generalized Huygens-Fresnel principle and the second moment theory of Wigner distribution function, the analytical expression of M2 factor of EEMGSM beams propagating in anisotropic turbulence was derived theoretically. Through numerical calculation and analysis, the variations of initial polarization, initial coherence, waist width, wavelength and turbulence structure constant with M2 factor were investigated. The results show that, M2 factor of EEMGSM beam decreases with the increase of order, wavelength and waist width. It decreases with the decrease of initial coherence and turbulence structure constant. Under the same conditions, the beam quality of EEMGSM beams with large initial polarization is less affected by anisotropy than that of EEMGSM beams with small initial polarization. Under the same conditions, the quality factor of EEMGSM beam is smaller than that of scalar elliptical Gaussian-Schell mode (EGSM) beam. EEMGSM beam has the advantage of alleviating the influence of anisotropic turbulence. The study has certain theoretical reference value for the research of free space optical communication.