We present a new method for fitting potential energy surfaces (PESs), which combines the compactness of analytical fits with the flexibility of interpolation. In essence, potentials are interpolated with potential-shaped interpolation functions. Numerically efficient global PESs are constructed from a few interpolation points, updated as new regions are sampled, and easily constrained by spectroscopic and thermodynamic data. As an application, we compute an interpolated potential surface for the state of thiophosgene, which combines CASSCF calculations with refinement based on experimental vibrational frequencies and tunneling splittings.