We measured carbon abundances and the $^{12}\mathrm{C}/^{13}\mathrm{C}$ ratio in 31 giant branch stars with previous CN and CH band measurements that span -2.33 < M$_{\rm V}$ < 0.18 in the globular cluster M10 (NGC 6254). Abundances were determined by comparing CO features at $∼2.3μm$ and specifically the $^{13}\mathrm{C}$O bandhead at $2.37μm$, to synthetic spectra generated with MOOG. The observed spectra were obtained with GNIRS on Gemini North with a resolution of R ≈3500. The carbon abundances derived from the IR spectra agree with previous [C/Fe] measurements found using CN and CH features at the near-UV/blue wavelength range. We found an average carbon isotope ratio of $^{12}\mathrm{C}/^{13}\mathrm{C}$ = 5.10$_{-0.17}^{+0.18}$ for first generation stars (CN-normal; 13 stars total) and $^{12}\mathrm{C}/^{13}\mathrm{C}$ = 4.84$_{-0.22}^{+0.27}$ for second generation stars (CN-enhanced; 15 stars). We therefore find no statistically significant difference in $^{12}\mathrm{C}/^{13}\mathrm{C}$ ratio between stars in either population for the observed magnitude range. Finally, we created models of the expected carbon, nitrogen, and $^{12}\mathrm{C}/^{13}\mathrm{C}$ surface abundance evolution on the red giant branch due to thermohaline mixing using the MESA stellar evolution code. The efficiency of the thermohaline mixing must be increased to a factor of ≈60 to match [C/Fe] abundances, and by a factor of ≈666 to match $^{12}\mathrm{C}/^{13}\mathrm{C}$ ratios. We could not simultaneously fit the evolution of both carbon and the $^{12}\mathrm{C}/^{13}\mathrm{C}$ ratio with models using the same thermohaline efficiency parameter.