In this Letter, we report the theoretical model and the experimental evidence of a mode-splitting cloning effect due to the resonant coupling between modes having different polarizations in weakly birefringent fiber Bragg grating (FBG) ring resonators. This modal coupling depends on the fiber birefringence and the FBG reflectivity. In the ideal case of the absence of birefringence, a single split-mode resonant structure can be observed in the resonator transmission spectrum due to the degeneracy removal of the two counter-propagating modes. In the presence of FBG birefringence, a secondary split doublet resulting in a clone of the initial one is generated. The described effect can be exploited for spectroscopic-sensing applications based on more complex split-mode dynamics.