Benchmarking and characterizing quantum states and logic gates is essential in the development of devices for quantum computing. We introduce a Bayesian approach to self-consistent process tomography, called fast Bayesian tomography (FBT), and experimentally demonstrate its performance in characterizing a two-qubit gate set on a silicon-based spin qubit device. FBT is built on an adaptive self-consistent linearization that is robust to model approximation errors. Our method offers several advantages over other self-consistent tomographic methods. Most notably, FBT can leverage prior information from randomized benchmarking (or other characterization measurements), and can be performed in real time, providing continuously updated estimates of full process matrices while data are acquired.