We define a logical framework that permits runtime verification to take place when a monitor has incomplete information about the underlying trace. A stateful access control proxy has the capacity to turn events into sets of possible events, resulting in what we call a ``multi-trace''. We describe a model of both proxy and monitor as extensions of Mealy machines, and provide an algorithm to lift a classical monitor into a sound, loss-tolerant monitor. Experiments on various scenarios show that the approach can account for various types of data degradation and access limitations, provides a tighter verdict than existing works in some cases, and preserves scalable performance with respect to trace length and model size.