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Predictive-quality computational modeling of heterogeneously catalyzed reactions has emerged as an important tool for the analysis and assessment of activity and activity trends. In contrast, more subtle selectivities and selectivity trends still pose a significant challenge to prevalent microkinetic modeling approaches that typically employ a mean-field approximation (MFA). Here, we focus on CO hydrogenation on Rh catalysts with the possible products methane, acetaldehyde, ethanol and water. This reaction has already been subject to a number of experimental and theoretical studies with conflicting views on the factors controlling activity and selectivity towards the more valuable higher oxygenates. Using accelerated first-principles kinetic Monte Carlo (KMC) simulations and explicitly and systematically accounting for adsorbate-adsorbate interactions through a cluster expansion approach, we model the reaction on the low-index Rh(111) and stepped Rh(211) surfaces. We find that the Rh(111) facet is selective towards methane, while the Rh(211) facet exhibits a similar selectivity towards methane and acetaldehyde. This is consistent with the experimental selectivity observed for larger, predominantly (111)-exposing Rh nanoparticles and resolves the discrepancy to earlier first-principles MFA microkinetic work that found the Rh(111) facet to be selective towards acetaldehyde. While the latter work tried to approximately account for lateral interactions through coverage-dependent rate expressions, our analysis demonstrates that this fails to sufficiently capture concomitant correlations among the adsorbed reaction intermediates that crucially determine the overall selectivity.