  We study the phase diagram of a topological string-net type lattice model in
the presence of geometrically frustrated interactions. These interactions drive
several phase transitions that reduce the topological order, leading to a rich
phase diagram including both Abelian ($\mathbb{Z}_2$) and non-Abelian
($\text{Ising}\times \overline{\text{Ising}}$) topologically ordered phases, as
well as phases with broken translational symmetry. Interestingly, one of these
phases simultaneously exhibits (Abelian) topological order and long-ranged
order due to translational symmetry breaking, with non-trivial interactions
between excitations in the topological order and defects in the long-ranged
order. We introduce a variety of effective models, valid along certain lines in
the phase diagram, which can be used to characterize both topological and
symmetry-breaking order in these phases, and in many cases allow us to
characterize the phase transitions that separate them. We use exact
diagonalization and high-order series expansion to study areas of the phase
diagram where these models break down, and to approximate the location of the
phase boundaries.
