The inverse method of proving the twistability of cross-spectral density (CSD) inevitably falls into spontaneous difficulties. Based on a nonnegative self-consistent design guideline for generating genuine CSDs introduced by Gori and Santarsiero, we demonstrate a feasible way for twisting partially coherent sources by sticking a Schell-model function to CSDs, which also determines the upper bound of the twisting strength. Analysis shows that the degree of coherence of a new class of twisted pseudo-Gaussian Schell-model beam is neither shift invariant nor shift-circular symmetric. In the presence of a vortex phase, the two different types of chiral phases affect each other and together control the propagation behavior. We further carry out an experiment to generate this non-uniformly correlated twisted beam using weighted superposition of mutually uncorrelated pseudo modes. The result is beneficial for devising nontrivial twisted beams and offers new opportunities.