Coprime sensor arrays (CSAs) can estimate the directions of arrival of O(MN) narrowband planewave sources using only O(M + N) sensors with the CSA product processor. All previous investigations on the product processed CSA's performance for detecting Gaussian signals assumed spatially white Gaussian noise. Considering the product and conventional delay-and-sum beamforming processors applied to the CSA geometry, this paper derives the detection gain for each processor under the deflection metric when the background noise is spatially correlated. The conditional probability distribution functions are also derived in closed form to evaluate and compare the receiver operating characteristic performances. Despite the nonlinear processing, the product processed CSA's detection performance closely rivals its conventional beamforming (CBF) counterpart, while also demonstrating greater robustness than the CBF to spatially correlated Gaussian noise.