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.
