Toroidal 3-manifolds have circularly orderable fundamental groups

In this article we prove that toroidal 3-manifolds have circularly-orderable fundamental groups by showing that they admit finite cyclic covers with left-orderable fundamental groups. These covers are also not L-spaces and often admit co-orientable taut foliations, as predicted by the L-space conjecture. As a consequence, we verify a conjecture of Ba and Clay, which characterises graph manifolds with circularly-orderable fundamental groups.