We investigate the ground-state phase structure of tilted dipolar bosons in a two-dimensional square optical lattice in the presence of cavity-mediated global interactions. Our analysis is based primarily on the single-site Gutzwiller mean-field (SGMF) method, supplemented by representative $2\times 2$ cluster Gutzwiller mean-field (CGMF) calculations. A momentum-space analysis shows that the cavity-mediated interaction selectively lowers the checkerboard (CB) ordering channel, while the stripe (ST) channel receives no direct cavity contribution, thereby rendering the CB-ST crossing angle tunable. For $U_l=0$, dipolar anisotropy drives a crossover from CB-dominated to ST-dominated ordering at ${\theta }_c\simeq 27.3^{\circ }$, with a periodic mixed-order supersolid (MSS) regime exhibiting simultaneously resolved CB and ST modulations near the crossing. The ground-state phase diagrams and the calculated compressibility curves $\kappa \left(\mu \right)$ further reveal several fractional density plateaus accompanied by $\kappa \simeq 0$, supporting their incompressible character within the single-site Gutzwiller mean field; among them, the $\overline{\rho }=1/8$ and $1/4$ plateaus are comparatively wide. For finite $U_l$, the channel-selective cavity contribution shifts the crossing angle, expands CB-ordered regions, and modifies the parameter ranges of the fractional incompressible and MSS regimes. Representative CGMF calculations retain the principal ordered regions obtained within the single-site Gutzwiller mean field, although their phase boundaries are quantitatively shifted. These results quantitatively demonstrate that the cavity-mediated interaction provides an independent control parameter for shifting the balance between CB and ST ordering tendencies and reorganizing the associated phase diagram. More broadly, the selective tunability of the cavity-coupled ordering channel provides a possible route to controlling competing ordering tendencies in systems with multiple long-range interactions.

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• Cold gases in optical lattices
• Dipolar gases
• Long-range interactions