  Hexagonal-to-orthorhombic structural transitions in RAl$_3$C$_3$ have been
studied by X-ray diffraction for R=Yb, Tm, Er, Dy, and Lu. We report how
orthorhombic domains are controlled by adjusting external magnetic fields. It
is shown that orthorhombic domains in RAl$_3$C$_3$ can be aligned by the field
when the R ion is magnetic. It is also shown that a single-domain state can be
switched to another single-domain state by the field even in paramagnetic
states with small induced moments. For R=Lu, however, it was not possible to
change the multidomain state by a field of up to 70 kOe. We discuss that domain
selection and switching are caused by magnetic anisotropies in the orthorhombic
phase of RAl$_3$C$_3$. We also propose an idea to explain the field-induced
antiferromagnetism revealed by Khalyavin et al. [Phys. Rev. B \textbf{87},
220406 (2013)].
