In recent years, the partially coherent beam carrying the twisted phase has received extensive attention due to its unique properties. A new partially coherent vector beam endowed with a twisted phase and a special spatial correlation structure is introduced, which is radially polarized twisted rectangular multi-Gaussian Schell-model beam. The cross-spectral density matrix elements of such beam passing through the ABCD optical system were derived, and the evolution properties, such as the normalized intensity distribution, the spectral degree of coherence (SDOC), and the spectral degree of polarization (SDOP) were investigated in detail. Numerical results show that the intensity distribution of the radially polarized rectangular multi-Gauss Schell-model beam without the twist phase gradually evolves from the hollow ring profile in the source plane into the rectangular flat-top profile in the focal plane. In comparison, the twisted phase carried by the radially polarized twisted rectangular multi-Gaussian Schell-model beam will not only induce the rotation of the beam spot, but also cause a series of changes in the SDOC and SDOP of the beam during the propagation process. The research results provide some theoretical guidance for the control of new vector structured beams with twisted phases, and have potential applications in beam shaping, optical micro-manipulation, and free space optical communications.