As data shift or new data become available, updating clinical machine learning models may be necessary to maintain or improve performance over time. However, updating a model can introduce compatibility issues when the behavior of the updated model does not align with user expectations, resulting in poor user-model team performance. Existing compatibility measures depend on model decision thresholds, limiting their applicability in settings where models are used to generate rankings based on estimated risk. To address this limitation, we propose a novel rank-based compatibility measure, $C^R$, and a new loss function that aims to optimize discriminative performance while encouraging good compatibility. Applied to a case study in mortality risk stratification leveraging data from MIMIC, our approach yields more compatible models while maintaining discriminative performance compared to existing model selection techniques, with an increase in $C^R$ of $0.019$ ($95\%$ confidence interval: $0.005$, $0.035$). This work provides new tools to analyze and update risk stratification models used in clinical care.