This paper shows that imaging the interior of solid bodies with fully non‐linear physics can be highly beneficial compared to imaging with the equivalent linearised tomographic methods, and that this is true for a variety of different types of physics. Including full non‐linearity provides interpretable uncertainties, and far greater depth of image penetration into unknown targets such as the Earth's subsurface. We use an adaptively parameterised Monte Carlo method to invert electrical resistivity data for the conductivity structure of the Earth, and demonstrate the method on two field datasets. Key results include the observation of directly interpretable uncertainty loops which define possible geometrical variations in the edges of isolated anomalies, hence quantifying the spatial resolution of these boundaries. These topologies of uncertainties are similar to those observed when performing fully non‐linear seismic travel time tomography. This shows that loop‐like uncertainty topologies are expected in the solutions to a wide variety of tomographic problems, using a variety of data types and hence laws of physics (here the Laplace equation; in previous work the Eikonal or ray equations). We also show that the depth to which we can construct a tomographic image using electrical data is extended by up to a factor of 8 using non‐linear methods compared to linearised inversion using common standard linearised programs. These advantages come at the cost of significantly increased computation. All of these results are illustrated on both synthetic and real data examples.