Reduced Order Models (ROMs) are indispensable tools for facilitating real-time digital twinning by simplifying complex systems into computationally manageable forms. By leveraging established physics-based approaches, ROMs gain a physical connotation, which is crucial for extrapolation and generalization tasks. The evaluation of ROMs requires a balance between computational efficiency and prediction accuracy relative to full-order models; a challenging trade-off given that ROM outputs support critical engineering decisions, such as those related to operation and maintenance. In this study, we propose an adaptive approach for selecting the optimal ROM, which adjusts to the evolving nature of monitored systems. Leveraging real-time monitoring data, we address the challenge of adaptive ROM Model selection. A Bayesian inference framework is developed for quantifying the posterior uncertainty associated with the resolution of each considered candidate ROM. Subsequently, we set up a decision-theoretic objective function for the ROM selection problem. The objective is to select the ROM resolution that maximizes the expected utility, which is a function of two competing attributes: (i) ROM precision, which accounts for the impact of deviations in predicted quantities on decision quality, and (ii) computational efficiency, to ensure feasibility in real-time applications.