Context. Most long gamma-ray bursts (GRBs) exhibit afterglows that are broadly consistent with external forward-shock emission, which is typically described by smooth broken power-law decays in the multiband light curve. However, a minority of well-sampled GRBs deviates from this behavior, including the GRB we investigated, GRB 250129A. This source shows multiple late-time rebrightenings at X-ray and optical wavelengths. Rebrightenings are often attributed to energy injection from prolonged activity of the central engine, refreshed shocks from delayed shell collisions, density jumps in the ambient medium, or to the angular jet structure and viewing-angle effects. Aims After a comprehensive analysis of the prompt emission of the GRB observed in gamma rays and in the near-infrared, we investigated the physical origin of the multiple X-ray and optical flaring episodes observed in GRB 250129A. Methods. We conducted comprehensive temporal and spectral multiband analyses of GRB 250129A. The physical processes at play in the afterglow light curves were investigated using several methods, ranging from empirical fitting to Bayesian inference. The high-quality monitoring of the flare episodes, together with the connection between the prompt emission and the afterglow, enabled us to test the consistency of the fireball model and to constrain, reject, or confirm alternative scenarios. Results. GRB 250129A is an interesting GRB with multiwavelength prompt and afterglow emission. By conducting the prompt and time-resolved analyses, we obtained an isotropic-equivalent energy of Eiso, γ = (1.35 ± 0.12)×1053 erg. By modeling the afterglow using an agnostic Bayesian framework (NMMA), we ruled out a single external-shock evolution and a one-time energy-injection scenario. By further performing numerical calculations, we demonstrated that the rebrightening episodes are consistent with refreshed shocks arising from delayed collisions between relativistic shells. This agrees with evolving outflow dynamics. Conclusions. Based on the consistency between our analyses of the prompt and afterglow GRB 250129A data, together with prior general knowledge on microphysical properties, we find that two statistically significant rebrightening episodes occurred within 1.1 days post trigger and can be explained by a sequence of refreshed shocks. We stress that the availability of temporally and spectrally rich GRB datasets, such as the one we present here, provides a powerful means to test current modeling frameworks.