The increasing availability of monitoring data from civil infrastructures has enabled the development of digital twins as effective tools for structural assessment and maintenance. In this context, physics-based models play a crucial role by providing interpretable and extrapolative predictions that support well-informed decision making. However, the creation of such models necessarily involves numerous modelling choices, ranging from simplifications of geometry and material behaviour to boundary and initial condition assumptions, which introduce model form uncertainty (MFU). This form of uncertainty manifests as unavoidable discrepancies between model predictions and reality, even after calibration, and must therefore be explicitly quantified. In this contribution, we present a framework for the calibration of simulation models under unavoidable discrepancies, using the Nibelungenbrücke in Worms (Germany) as a case study. The bridge is equipped with a monitoring system that provides real-time thermal data, which we use to calibrate a transient thermal model serving as the physics-based component of the digital twin. A key challenge arises from the fact that the model setup, particularly the definition of initial conditions, boundary conditions, and thermal loads, is itself affected by MFU. To address this, we adopt an embedded approach in which MFU is treated through stochastic extensions of selected model parameters, including a representation of the uncertainty in the boundary conditions. This formulation allows the uncertainty associated with modelling assumptions to be integrated within the calibration rather than treated as an external bias. The embedded approach enables the systematic propagation of MFU through the model response to quantities of interest, thereby providing not only uncertainty bounds on predictions but also diagnostic insight into the validity of the modelling assumptions themselves. The resulting workflow offers a robust path toward trustworthy digital twins, capable of accommodating structural monitoring data while maintaining transparency about the limits of the underlying physical model.