We use the Ginzburg-Landau theory for a chiral $p$-wave superconductor to describe the filamentary superconducting phase nucleating at temperatures higher than the bulk transition temperature near the interfaces between Ru metal and ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ in eutectic ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$-Ru. The peculiar phase diagram of magnetic field versus temperature shows very distinct properties for different directions of the magnetic field. A clear feature of the magnetic fields in the basal plane of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$, perpendicular to the chiral axis ($c$ axis), is the occurrence of a second superconducting transition with a new order parameter visible as a zero-bias anomaly in tunneling spectroscopy. For the field along the $c$ axis of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$, two transitions merge, indicating that the field drives the second-order parameter through the paramagnetic coupling to a chiral supercurrent. The resulting phenomenology shows that the assumption of a chiral superconducting order parameter in ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ yields an image reproducing the qualitative properties observed in the experiment. This discussion is qualitatively analogous to the corresponding chiral even- and odd-parity channels.