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Interferometric observables are strongly correlated, yet it is common practice to ignore these correlations in the data analysis process. We develop an empirical model for the correlations present in Very Large Telescope Interferometer GRAVITY data and show that properly accounting for them yields fainter detection limits and increases the reliability of potential detections. We extracted the correlations of the (squared) visibility amplitudes and the closure phases directly from intermediate products of the GRAVITY data reduction pipeline and fitted our empirical models to them. Then, we performed model fitting and companion injection and recovery tests with both simulated and real GRAVITY data, which are affected by correlated noise, and compared the results when ignoring the correlations and when properly accounting for them with our empirical models. When accounting for the correlations, the faint source detection limits improve by a factor of up to $\sim 2$ at angular separations $> 20~\rm{mas}$. For commonly used detection criteria based on $χ^2$ statistics, this mostly results in claimed detections being more reliable. Ignoring the correlations present in interferometric data is a dangerous assumption which might lead to a large number of false detections. The commonly used detection criteria (e.g. in the model fitting pipeline CANDID) are only reliable when properly accounting for the correlations; furthermore, instrument teams should work on providing full covariance matrices instead of statistically independent error bars as part of the official data reduction pipelines.