Reconstructing the structure of thin films and multilayers from measurements of scattered x-rays or neutrons is key to progress in physics, chemistry, and biology. However, finding all structures compatible with reflectometry data is computationally prohibitive for standard algorithms, which typically results in unreliable analysis with only a single potential solution identified. We address this lack of reliability with a probabilistic deep learning method that identifies all realistic structures in seconds, redefining standards in reflectometry. Our method, prior-amortized neural posterior estimation (PANPE), combines simulation-based inference with adaptive priors that inform the inference network about known structural properties and controllable experimental conditions. PANPE networks support key scenarios such as high-throughput sample characterization, real-time monitoring of evolving structures, or the corefinement of several experimental datasets and can be adapted to provide fast, reliable, and flexible inference across many other inverse problems.