Not yet reviewed by Pith; the record is open.

A confident discovery of physics beyond what has been consistently modeled from gravitational wave (GW) data requires a technique that can distinguish between noise artifacts and unmodeled signatures while also shedding light on the underlying physics. We propose a new data analysis method, \texttt{SCoRe} (Structured Correlated Residual), to search for unmodeled physics in the GW data which can cover both of these aspects. The method searches for structure in the cross-correlation power spectrum of the residual strain between pairs of GW detectors. It does so by projecting this power spectrum onto a frequency-dependent template. The template may be model-independent or model-dependent and is constructed based on the properties of the GW source parameters. The projection of the residual strain enables the distinction between noise artifacts and any true signal while capturing possible dependence on the GW source parameters. Our method is constructed in a Bayesian framework and we have shown its application on a model-independent toy example and for a model motivated by an effective field theory of gravity. The method developed here will be useful to search for a large variety of new physics and yet-to-be-modeled known physics in the GW data accessible from the current network of LIGO-Virgo-KAGRA detectors and from future earth- and space-based GW detectors such as A+, LISA, Cosmic Explorer, and Einstein Telescope.