Calibrating for direction-dependent ionospheric distortions in visibility data is one of the main technical challenges that must be overcome to advance low-frequency radio astronomy. In this paper, we propose a novel probabilistic, tomographic approach that utilises Gaussian processes to calibrate direction-dependent ionospheric phase distortions in low-frequency interferometric data. We suggest that the ionospheric free electron density can be modelled to good approximation by a Gaussian process restricted to a thick single layer, and show that under this assumption the differential total electron content must also be a Gaussian process. We perform a comparison with a number of other widely successful Gaussian processes on simulated differential total electron contents over a wide range of experimental conditions, and find that, in all experimental conditions, our model is better able to represent observed data and generalise to unseen data. The mean equivalent source shift imposed by our predictive errors are half as large as those of the best competitor model. We find that it is possible to partially constrain the hyperparameters of the ionosphere from sparse-and-noisy observed data. Our model provides an alternative explanation for observed phase structure functions deviating from Kolmogorov’s five-thirds turbulence, turnover at high baselines, and diffractive scale anisotropy. We show that our model performs tomography of the free electron density both implicitly and cheaply. Moreover, we find that even a fast, low-resolution approximation of our model yields better results than the best alternative Gaussian process, implying that the geometric coupling between directions and antennae is a powerful prior that should not be ignored.