We present COREA, the first unified three-tasks framework that couples an SDF and relightable 3D Gaussians (3DGS) to jointly support SH-based novel-view synthesis (NVS), surface reconstruction, and inverse physically-based rendering (inverse PBR). While recent relightable 3DGS methods have progressed, inverse PBR remains bottlenecked by normal estimation, as the discrete nature of 3DGS often yields oversmoothed and unstable normals. To address this limitation, COREA couples the complementary geometric properties of an SDF and relightable 3DGS on a shared underlying surface, where geometry-constrained relightable 3DGS provides reliable depth signals to anchor SDF geometry and the continuous SDF normal field provides spatially consistent supervision for Gaussian normal learning. We couple these signals through depth-guided alignment and normal supervision with normal-aware densification, and introduce Dual-Density Control to regulate densification by balancing photometric and geometric gradients for stable, memory-efficient training. Experiments on standard benchmarks show that COREA is the only framework that supports all three tasks, achieving competitive performance overall, with particularly superior results in inverse PBR.