  Motivated by conjectures relating group orderability, Floer homology, and
taut foliations, we discuss a systematic and broadly applicable technique for
constructing left-orders on the fundamental groups of rational homology
3-spheres. Specifically, for a compact 3-manifold $M$ with torus boundary, we
give several criteria which imply that whole intervals of Dehn fillings of $M$
have left-orderable fundamental groups. Our technique uses certain
representations from $\pi_1(M)$ into $\widetilde{\mathrm{PSL}_2 \mathbb{R}}$,
which we organize into an infinite graph in $H^1(\partial M; \mathbb{R})$
called the translation extension locus. We include many plots of such loci
which inform the proofs of our main results and suggest interesting avenues for
future research.
