Honestly, it's been a couple of days and I don't know what exactly I was on about when I should have been sleeping, except probably making the point that the Bo

Honestly, it's been a couple of days and I don't know what exactly I was on about when I should have been sleeping, except probably making the point that the Bourgoin et al paper was about the SME -- which is after all a modification of GR and the Standard Model in which it's normal to quantify arbitrary departures from Lorentz and CPT invariance -- rather than GR sensu strictissimo. An SME model that preserves local Lorentz invariance everywhere that GR does (or at least, everywhere in the EFT limits) does not necessarily carry a Lorentzian manifold in its mathematical structure (e.g. LIFs might arise due to suppressed LIV effects in a more fundamental theory), and conversely, an SME model that does have a Lorentzian manifold background might still have some Lorentz invariance breaking term. I think we're all in agreement but talking past each other in English a bit when it comes to GR itself.