Context. Several strong solar resonance lines show observable linear scattering polarization signals, holding a great potential for investigating the magnetism of the outer solar atmosphere. Accurately modeling these signals requires consideration of the detailed 3D structure of the solar atmosphere and accounting for partial frequency redistribution (PRD) effects in scattering processes. This setting has so far been computationally inaccessible. Aims. We aim to present the first scientific application of the TRIP code, which is a novel software for modeling the polarization of strong resonance lines in comprehensive 3D models of the solar atmosphere, accounting for PRD effects. We aim to verify the code and explore the combined action of magnetic fields, plasma bulk velocities, PRD effects, and the 3D structure of the solar atmosphere on scattering polarization. Methods. We ran TRIP to synthesize the Stokes profiles of the Ca I line at 4227 Å, accounting for angle-dependent PRD effects, in a 3D atmospheric model extracted from a radiation-magneto-hydrodynamic simulation. We effectively solved the resulting large-scale radiative transfer problem iteratively and in parallel using the flexible Generalized Minimal Residual (FGMRES) method with a physics-based preconditioner. Results. We successfully cross-checked TRIP with the established PORTA code, modeling scattering processes in the limit of complete frequency redistribution (CRD). We analyzed the relative impact of PRD effects, horizontal atmospheric inhomogeneities, magnetic fields, and bulk velocities on the polarization of the radiation emergent at the disk center, finding that bulk velocities are a dominant factor, while magnetic fields play a comparatively minor role. Conclusions. TRIP has been successfully tested and is ready for large-scale applications in state-of-the-art, 3D atmospheric models, allowing quantitative comparisons between synthetic and observational spectropolarimetric data of strong resonance lines.