A digital twin (DT) contains a set of virtual models of real systems and processes that are synchronized with their physical counterparts. In a setup in which contact with the physical world is maintained through sensors and actuators that are wirelessly connected to the DT's computing engine, DT alignment requires periodic status updates, while safety-critical messages and fault conditions call for low-latency anomaly reporting, creating a fundamental trade-off in how wireless resources are used. We present a medium access framework combining pull-based updates, centrally scheduled according to goal-oriented principles, with urgent push-based updates, for which transmission decisions are made directly by the sensors. This enables the system to quickly detect and recover from anomalies while maintaining DT alignment. We thus design a push-pull scheduler (PPS) that strikes a balance in the trade-off between DT alignment in normal conditions and anomaly reporting, optimizing resource usage and reducing DT drift by 20 - 30% with respect to state-of-the-art solutions while maintaining the same anomaly detection guarantees, or reducing worst-case anomaly detection times by 30 - 70% while meeting the same DT alignment conditions.

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