  In this paper, we develop a method for constructing left-orders on the
fundamental groups of rational homology 3-spheres. We begin by constructing the
holonomy extension locus of a rational homology solid torus $M$, which encodes
the information about peripherally hyperbolic
$\widetilde{\text{PSL}_2\mathbb{R}}$ representations of $\pi_1(M)$. Plots of
the holonomy extension loci of many rational homology solid tori are shown, and
the relation to left-orderability is hinted. Using holonomy extension loci, we
study rational homology 3-spheres coming from Dehn filling on rational homology
solid tori and construct intervals of Dehn fillings with left-orderable
fundamental group.
