  The presence of domains in ferroic materials can negatively affect their
macroscopic properties and hence their usefulness in device applications. From
an experimental perspective, measuring materials comprising multiple domains
can complicate the interpretation of material properties and their underlying
mechanisms. In general, BiFeO$_3$ films tend to grow with multiple magnetic
domains and often contain multiple ferroelectric and ferroelastic domain
variants. By growing (111)-oriented BiFeO$_3$ films on an orthorhombic
TbScO$_3$ substrate, we are able to overcome this, and, by exploiting the
magnetoelastic coupling between the magnetic and crystal structures, bias the
growth of a given magnetic-, ferroelectric-, and structural-domain film. We
further demonstrate the coupling of the magnetic structure to the ferroelectric
polarisation by showing the magnetic polarity in this domain is inverted upon
180$^\circ$ ferroelectric switching
