  The analysis of strong lensing images usually involves an external
convergence and shear, which are meant to model the effect of perturbations
along the line of sight, on top of the main lens. Such a description of
line-of-sight perturbations supposes that the corresponding gravitational
fields can be treated in the tidal regime. Going one step further introduces
additional effects, known as flexion, which have been hitherto neglected in
strong lensing. In this work, we build a minimal model for the line-of-sight
flexion, which adds four new complex parameters to the lens model. Contrary to
convergence and shear, the line-of-sight flexion cannot be projected onto the
main lens plane. For a $\Lambda$CDM cosmology, we predict the typical
line-of-sight flexion to be on the order of $10^{-3} \mathrm{arcsec}^{-1}$ on
galactic scales. Neglecting its effect in lens modelling is found to bias the
recovery of other parameters; in particular, the line-of-sight shear can be
biased up to $2\sigma$. Accounting for the line-of-sight flexion in our minimal
framework restores accuracy, at the the cost of degrading precision. With
current imaging capabilities, the line-of-sight flexion is unlikely to be
measurable on individual strong lensing images; it must therefore be considered
a nuisance parameter rather than an observable in its own right.
