  The ill-posedness of Calder\'on's inverse conductivity problem, responsible
for the poor spatial resolution of Electrical Impedance Tomography (EIT), has
been an impetus for the development of hybrid imaging techniques, which
compensate for this lack of resolution by coupling with a second type of
physical wave, typically modeled by a hyperbolic PDE. Here we show how, using
EIT data alone, to efficiently detect interior jumps and other singularities of
the conductivity. Analysis of the complex geometrical optics solutions of
Astala and P\"aiv\"arinta [\emph{Ann. Math.}, {\bf 163} (2006)] in 2D makes it
possible to exploit an underlying complex principal type structure of the
problem. We show that the leading term in a Neumann series is an invertible
nonlinear generalized Radon transform of the conductivity. The wave front set
of all higher-order terms can be characterized, and, under a prior, some are
smoother than the leading term. Numerics indicate that this approach
effectively detects inclusions within inclusions via EIT.
