  We generalise work of Young-Eun Choi to the setting of ideal triangulations
with vertex links of arbitrary genus, showing that the set of all (possibly
incomplete) hyperbolic cone-manifold structures realised by positively oriented
hyperbolic ideal tetrahedra on a given topological ideal triangulation and with
prescribed cone angles at all edges is (if non-empty) a smooth complex manifold
of dimension the sum of the genera of the vertex links. Moreover, we show that
the complex lengths of a collection of peripheral elements give a local
holomorphic parameterisation of this manifold.
