  We show that the observed time-reversal symmetry breaking (TRSB) of the
superconducting state in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ can be
understood as originating from inhomogeneous strain fields near edge
dislocations of the crystal. Specifically, we argue that, without strain
inhomogeneities, $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ is a
single-component, time-reversal symmetric superconductor, likely with
$d_{x^{2}-y^{2}}$ symmetry. However, due to the strong strain inhomogeneities
generated by dislocations, a slowly-decaying sub-leading pairing state
contributes to the condensate in significant portions of the sample. As it
phase winds around the dislocation, time-reversal symmetry is locally broken.
Global phase locking and TRSB occur at a sharp Ising transition that is not
accompanied by a change of the single-particle gap and yields a very small heat
capacity anomaly. Our model thus explains the puzzling absence of a measurable
heat capacity anomaly at the TRSB transition in strained samples, and the
dilute nature of the time-reversal symmetry broken state probed by muon spin
rotation experiments. We propose that plastic deformations of the material may
be used to manipulate the onset of broken time-reversal symmetry.
