  We investigate the phase diagram of dirty two-band superconductors. This
paper primarily focuses on the properties and observability of the
time-reversal symmetry-breaking $s+is$ superconducting states, which can be
generated in two-band superconductors by interband impurity scattering. We show
that such states can appear in two distinct ways. First, according to a
previously discussed scenario, the $s+is$ state can form as an intermediate
phase at the impurity-driven crossover between $s_{\pm}$ and $s_{++}$ states.
We show that there is a second scenario where domains of the $s+is$ state
exists in the form of an isolated dome inside the $s_{\pm}$ domain, completely
detached from the transition between $s_{\pm}$ and $s_{++}$ states. We
demonstrate that in both cases the $s+is$ state, generated by impurity
scattering exists in an extremely small interval of impurity concentrations.
Although this likely precludes direct experimental observation of the $s+is$
state formation due to this mechanism, this physics leads to the appearance of
a region inside both the $s_{\pm}$ and $s_{++}$ domains with unusual properties
due to softening of normal modes.
