Strongly nonlocal nonlinear media (SNNM) can effectively suppress beam collapse during propagation, and investigating the evolution of optical fields in such media is of great significance for the stable control of multidimensional spatial solitons. However, for partially coherent optical fields carrying both twist phase and cross phase, the underlying mechanism of their joint modulation in SNNM remains to be further clarified. In this work, a twist- and cross-phase-modulated Gaussian-Schell model beam is constructed. Based on the Snyder-Mitchell model and the Collins formula, analytical expressions for the cross-spectral density function, spectral density, and spectral degree of coherence(SDOC) after propagation in SNNM are derived. Numerical calculations are then performed to analyze the beam evolution under different modulation parameters. The results show that the beam can propagate in a stable soliton state when diffraction is balanced by nonlinear self-focusing, whereas periodic breathing behavior occurs when this balance is broken. The cross phase mainly governs the anisotropy and rotational characteristics of the spectral density and coherence structure, while the twist phase primarily affects the breathing scale and the spatial broadening of the orbital angular momentum (OAM) flux density. Their combined modulation enables more precise control of the propagation dynamics of the beam. These results enrich the theoretical study of propagation control and optical manipulation of partially coherent optical fields in strongly nonlocal nonlinear media.