  Supersymmetric bosonic backgrounds governed by first-order BPS equations, can
be realised in a much broader setting by relaxing the requirement of closure of
the superalgebra beyond the level of quadratic fermion terms. The resulting
pseudo-supersymmetric theories can be defined in arbitrary spacetime
dimensions. We focus here on the ${\cal N}=1$ pseudo-supersymmetric extensions
of the arbitrary-dimensional bosonic string action, which were constructed a
few years ago. In this paper, we recast these in the language of generalised
geometry. More precisely, we construct the action and the corresponding
supersymmetry transformation rules in terms of O($D$)$\times$O($D$) covariant
derivatives, and we discuss consistent truncations on manifolds with
generalised $G$-structure. As explicit examples, we discuss Minkowski$\times G$
vacuum solutions and their corresponding pseudo-supersymmetry. We also briefly
discuss squashed group manifold solutions, including an example with a
Lorentzian signature metric on the group manifold $G$.
