In recent years, the growing complexity in data collection technologies has lead to a variety of models used to accommodate tensor-valued data. However, when the data-generating mechanism is sensible to the operator (or "technician") who collected the data, the error-variability attributed to this gauge must be studied separately from the "pure error" that originates from the model. In this article we introduce a tensor-response two-way random-effects model that quantifies the replicability and reproducibility of a high-dimensional measurement process. We provide an expectation-maximization algorithm to estimate the parameters involved in our model, and provide statistics that naturally generalize those used in univariate and multivariate Gauge R&R studies. We apply our methodology to a matrix-response measurement process that originates from matching fractured surfaces, which has implications in forensic sciences to decide whether two fractured surfaces belong to the same knife. We also apply the methodology to a tensor-response model of functional magnetic resonance imaging, where several subjects were exposed to the same stimuli.