Many applications of tomography seek to image two-phase materials, such as oil and air, with the idealized aim of producing a binary reconstruction. The method of Tamburrino and Rubinacci ( 2002 Inverse Problems 18 1809-29) provides a non-iterative approach, which requires modest computational effort, and hence appears to achieve this aim. Specifically, it requires the solution of a number of forward problems increasing only linearly with the number of elements used to represent the domain where the resistivity is unknown. However, even when low measurement noise is present it may be that not all domain elements can be classified and hence only a partial reconstruction is possible. This paper looks at the use of a Bayesian approach based on the monotonicity information for reconstructing the shape of a homogeneous inclusion in another homogeneous material. In particular, the monotonicity criterion is used to fix the resistivity of some pixels. The uncertain pixel resistivities are then estimated, conditional upon the fixed values. This has the effect of both producing better reconstructions and reducing the computational burden by up to an order of magnitude in the examples considered. The methods are illustrated using simulation examples covering a range of object geometries.