An essential part of any analysis using radio detector data is robust and accurate reconstruction of the physical parameters governing the observed signals. However, most current reconstruction methods ignore bin-to-bin noise correlations, which limits reconstruction resolution and prevents reliable event-by-event uncertainty estimates. In this contribution, we present a likelihood-based approach to reconstruction that correctly models the band-limited/correlated nature of radio detector noise. This enables correct event-by-event uncertainty estimates, improves reconstruction accuracy, and has applications in detector optimization. We demonstrate the method on simulated in-ice neutrino and cosmic-ray air-shower events. The reconstruction code is available through the open-source software NuRadioReco.