We put forward a novel formulation of the vortex gas model of turbulent circulation statistics to address the challenging case of nonplanar circulation contours. Relying upon a field-theoretical description, statistical moments of the circulation turn out to be functionally dependent on specific optimal surfaces bounded by the circulation loops. Circulation is modeled in the optimal curved spaces with the help of scalar vertex operators that represent the multifractal density fluctuations of Gaussian-correlated vortex structures. We show that minimal surfaces are optimal within the inertial range, but subdominant deviations are expected to become significant for contours with linear dimensions close to the Kolmogorov dissipation length. As a case study, we demonstrate the model’s applicability through a Monte Carlo evaluation of the circulation probability distribution function for nonplanar contours, which is in excellent agreement with results from extensive direct numerical simulations.