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The ability to control materials properties through interface engineering is demonstrated by the appearance of conductivity at the interface of certain insulators, most famously the {001} interface of the band insulators LaAlO$_{3}$ and TiO$_{2}$-terminated SrTiO$_{3}$ (STO). Transport and other measurements in this system show a plethora of diverse physical phenomena. To better understand the interface conductivity, we used scanning superconducting quantum interference device microscopy to image the magnetic field locally generated by current in an interface. At low temperature, we found that the current flowed in conductive narrow paths oriented along the crystallographic axes, embedded in a less conductive background. The configuration of these paths changed on thermal cycling above the STO cubic-to-tetragonal structural transition temperature, implying that the local conductivity is strongly modified by the STO tetragonal domain structure. The interplay between substrate domains and the interface provides an additional mechanism for understanding and controlling the behaviour of heterostructures.