The posterior distribution of earth models that fit observed geophysical data convey information on the uncertainty with which they are resolved. From another perspective, the non-uniqueness inherent in most geophysical inverse problems of interest can be quantified by examining the posterior model distribution converged upon by a Bayesian inversion. In this work we apply a reversible jump Markov chain Monte Carlo method to sample the posterior model distribution for the anisotropic 1-D seafloor conductivity constrained by marine controlled source electromagnetic data. Unlike conventional gradient based inversion approaches, our algorithm does not require any subjective choice of regularization parameter, and it is self parametrizing and trans-dimensional in that the number of interfaces with a resistivity contrast at depth is variable, as are their positions. A synthetic example demonstrates how the algorithm can be used to appraise the resolution capabilities of various electromagnetic field components for mapping a thin resistive reservoir buried beneath anisotropic conductive sediments. A second example applies the method to survey data collected over the Pluto gas field on the Northwest Australian shelf. A benefit of our Bayesian approach is that subsets of the posterior model probabilities can be selected to test various hypotheses about the model structure, without requiring further inversions. As examples, the subset of model probabilities can be viewed for models only containing a certain number of layers, or for models where resistive layers are present between a certain interval as suggested by other geological constraints such as seismic stratigraphy or nearby well logs.