Large spatial data sets now record many correlated variables at many thousands of locations, often on domains where Euclidean distance misrepresents proximity. The central difficulty is modelling the cross-variable dependence jointly while retaining variable-level interpretation. We introduce the bigraphical Mat'ern-Whittle process, a multivariate Gaussian process that resolves this with two graphs. A spatial graph generates the Mat'ern structure of each variable through a fractional power of a graph Laplacian, so the process is valid on any topology, with per-variable range, smoothness and amplitude. A directed acyclic variable graph encodes the scientific structure: we prove that each absent edge yields an exact conditional independence between the corresponding fields. We further prove that the operator determinant does not involve the cross-dependence coefficients, which keeps matrix-free likelihood evaluation and Bayesian learning of the variable graph tractable at scale. Estimation requires only sparse matrix-vector products and scales to tens of millions of space-variable pairs. In simulations the method recovered parameters and graphs accurately, remained robust under misspecification, and halved held-out prediction error on a non-convex domain. In a spatial transcriptomics section with 19,809 cells and 1,122 genes, fitted in 75 minutes on a laptop, borrowing across the learned gene graph reduced held-out prediction error by 50 to 91 percent. Theoretical challenges, such as the achievable efficiency of estimating the variance of the nugget, are also explored.

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