Dense and accurate depth estimation is essential for robotic manipulation, grasping, and navigation, yet currently available depth sensors are prone to errors on transparent, specular, and general non-Lambertian surfaces. To mitigate these errors, large-scale monocular depth estimation approaches provide strong structural priors, but their predictions can be potentially skewed or mis-scaled in metric units, limiting their direct use in robotics. Thus, in this work, we propose a training-free depth grounding framework that anchors monocular depth estimation priors from a depth foundation model in raw sensor depth through factor graph optimization. Our method performs a patch-wise affine alignment, locally grounding monocular predictions in metric real-world depth while preserving fine-grained geometric structure and discontinuities. To facilitate evaluation in challenging real-world conditions, we introduce a benchmark dataset with dense scene-wide ground truth depth in the presence of non-Lambertian objects. Ground truth is obtained via matte reflection spray and multi-camera fusion, overcoming the reliance on object-only CAD-based annotations used in prior datasets. Extensive evaluations across diverse sensors and domains demonstrate consistent improvements in depth performance without any (re-)training. We make our implementation publicly available at https://anchord.cs.uni-freiburg.de.