  String backgrounds of the form $\mathbb{M}_3 \times {\cal M}_7$ where
$\mathbb{M}_3$ denotes $3$-dimensional Minkowski space while ${\cal M}_7$ is a
$7$-dimensional G$_2$-manifold, are characterised by the property that the
world-sheet theory has a Shatashvili-Vafa (SV) chiral algebra. We study the
generalisation of this statement to backgrounds where the Minkowski factor
$\mathbb{M}_3$ is replaced by ${\rm AdS}_3$. We argue that in this case the
world-sheet theory is characterised by a certain ${\cal N}=1$ superconformal
${\cal W}$-algebra that has the same spin spectrum as the SV algebra and also
contains a tricritical Ising model ${\cal N}=1$ subalgebra. We determine the
allowed representations of this ${\cal W}$-algebra, and analyse to which extent
the special features of the SV algebra survive this generalisation.
