  In electrical impedance tomography, we aim to solve the conductivity within a
target body through electrical measurements made on the surface of the target.
This inverse conductivity problem is severely ill-posed, especially in real
applications with only partial boundary data available. Thus regularization has
to be introduced. Conventionally regularization promoting smooth features is
used, however, the Mumford--Shah regularizer familiar for image segmentation is
more appropriate for targets consisting of several distinct objects or
materials. It is, however, numerically challenging. We show theoretically
through $\Gamma$-convergence that a modification of the Ambrosio--Tortorelli
approximation of the Mumford--Shah regularizer is applicable to electrical
impedance tomography, in particular the complete electrode model of boundary
measurements. With numerical and experimental studies, we confirm that this
functional works in practice and produces higher quality results than typical
regularizations employed in electrical impedance tomography when the
conductivity of the target consists of distinct smoothly-varying regions.
