  Systems of interacting fermions can give rise to ground states whose
correlations become effectively free-fermion-like in the thermodynamic limit,
as shown by Baxter for a class of integrable models that include the
one-dimensional XYZ spin-$\frac{1}{2}$ chain. Here, we quantitatively analyse
this behaviour by establishing the relation between system size and correlation
length required for the fermionic Gaussianity to emerge. Importantly, we
demonstrate that this behaviour can be observed through the applicability of
Wick's theorem and thus it is experimentally accessible. To establish the
relevance of our results to possible experimental realisations of XYZ-related
models, we demonstrate that the emergent Gaussianity is insensitive to weak
variations in the range of interactions, coupling inhomogeneities and local
random potentials.
