  Quantum phase transitions between different topologically ordered phases
exhibit rich structures and are generically challenging to study in microscopic
lattice models. In this work, we propose a tensor-network solvable model that
allows us to tune between different symmetry enriched topological (SET) phases.
Concretely, we consider a decorated two-dimensional toric code model for which
the ground state can be expressed as a two-dimensional tensor-network state
with bond dimension $D=3$ and two tunable parameters. We find that the
time-reversal (TR) symmetric system exhibits three distinct phases (i) an SET
toric code phase in which anyons transform non-trivially under TR, (ii) a toric
code phase in which TR does not fractionalize, and (iii) a topologically
trivial phase that is adiabatically connected to a product state. We
characterize the different phases using the topological entanglement entropy
and a membrane order parameter that distinguishes the two SET phases. Along the
phase boundary between the SET toric code phase and the toric code phase, the
model has an enhanced $U(1)$ symmetry and the ground state is a quantum
critical loop gas wavefunction whose squared norm is equivalent to the
partition function of the classical $O(2)$ model. By duality transformations,
this tensor-network solvable model can also be used to describe transitions
between SET double-semion phases and between $\mathbb{Z}_2\times\mathbb{Z}_2^T$
symmetry protected topological phases in two dimensions.
