  The Heisenberg-Kitaev (HK) model on the triangular lattice is conceptually
interesting for its interplay of geometric and exchange frustration. HK models
are also thought to capture the essential physics of the spin-orbital
entanglement in effective $j=1/2$ Mott insulators studied in the context of
various 5d transition metal oxides. Here we argue that the recently synthesized
Ba$_3$IrTi$_2$O$_9$ is a prime candidate for a microscopic realization of the
triangular HK model. We establish that an infinitesimal Kitaev exchange
destabilizes the 120$^\circ$ order of the quantum Heisenberg model and results
in the formation of an extended $\mathbb{Z}_2$-vortex crystal phase in the
parameter regime most likely relevant to the real material. Using a combination
of analytical and numerical techniques we map out the entire phase diagram of
the model, which further includes various ordered phases as well as an extended
nematic phase around the antiferromagnetic Kitaev point.
