This paper considers the passive detection of a signal common to two multi-sensor arrays. We consider Gaussian received signals and noises with positive-definite, but otherwise unstructured covariance matrices. Under the null hypothesis, the composite covariance matrix for the two arrays is block-diagonal with arbitrary positive definite (PD) blocks, whereas under the alternative, it is modeled as an unstructured covariance matrix. Assuming complex inverse-Wishart priors for the unknown covariance matrices, the proposed test relies on the marginalized likelihood ratio, where the unknown parameters (i.e., the covariance matrices) are integrated out. A proper choice of hyper-parameters of the prior distribution shows that the Bayesian-inspired test reduces to a regularized canonical correlation analysis (CCA) detector. Simulation results show the superior performance of the proposed method compared to the generalized likelihood ratio test (GLRT), which is given by a function of the canonical correlations.