Future searches for a gravitational-wave background using Earth-based gravitational-wave detectors might be impacted by correlated noise sources. A well-known example are the Schumann resonances, which are extensively studied in the context of searches for a gravitational-wave background. Earlier work has shown that a technique termed “gravitational-wave geodesy” can be used to generically differentiate observations of a gravitational-wave background from signals due to correlated terrestrial effects, requiring true observations to be consistent with the known geometry of our detector network. The key result of this test is a Bayes factor between the hypotheses that a candidate signal is astrophysical or terrestrial in origin. Here, we further formalize the geodesy test, mapping distributions of false-alarm and false-acceptance probabilities to quantify the degree with which a given Bayes factor will boost or diminish our confidence in an apparent detection of the gravitational-wave background. To define the false alarm probability of a given Bayes factor, we must have knowledge of our null hypothesis: the space of all possible correlated terrestrial signals. Since we do not have this knowledge, we instead construct a generic space of smooth functions in the frequency domain using Gaussian processes, which we tailor to be conservative. This enables us to use draws from our Gaussian processes as a proxy for all possible nonastrophysical signals. During the O2 observing run, the LIGO and Virgo collaborations observed an $\mathrm{SNR}=1.25$ excess in their search for an isotropic gravitational-wave background. To demonstrate the utility of gravitational-wave geodesy, we apply the method to the observed cross-correlated data.