  We construct a new class of non-supersymmetric ten-dimensional type II flux
vacua, by studying first order differential equations which are deformations of
the $\mathcal{N}=1$ supersymmetry conditions. We do so within the context of
Generalised Complex Geometry, where there is a natural interpretation of the
$\mathcal{N}=1$ supersymmetry conditions in terms of calibration conditions for
probe D-branes, called D-string, domain-wall or space-filling branes, depending
on them wrapping two, three or four non-compact dimensions. We focus on the
class of non-supersymmetric vacua violating the D-string calibration condition,
and write down their general equations of motion in the language of pure
spinors. We solve them for a subclass of vacua, where the deformation of the
calibration condition is dictated by the foliated geometry of the internal
space. We also construct backgrounds violating both the domain-wall and
D-string calibration conditions, generalising the one-parameter DWSB class of
backgrounds introduced in L\"ust et al. We present several explicit solutions
with SU$(2)$ and SU$(3)$ structures, and we investigate briefly their
associated low energy effective theories.
