Monolayer nanosheets of zirconium phosphate in aqueous suspension exhibit short-range repulsion and long-range attraction, producing, at overall volume fractions larger than about half a percent, phase separation into higher-concentration liquid crystal and lower-concentration isotropic regions. At high concentrations, this phase separation takes the form of an emulsion of condensed, liquid-crystalline droplets, which anneal to form lens-shaped tactoids. These tactoids provide an opportunity to study the liquid crystal ordering of inorganic nanosheets in the limit of large shape anisotropy (diameter/thickness~400) and high packing fraction (volume fraction ≳ 70%). The internal liquid crystal structure of the tactoids remains nematic even under conditions that would usually favor ordering into lamellar smectics. Local lamellar ordering is suggested by short-range, smectic-like layer correlations, but a full transition into a smectic phase appears to be inhibited by the nanosheet edges, which act as a perturbative population of dislocation loops in the system of layers. Under conditions of thermal equilibrium, the nanoplates organize positionally to enable bend deformation of the director, a hallmark of the nematic phase and its principal distinction from the smectic, where bend must be expelled.