  We study a $(2+1)$-dimensional stochastic interface growth model, that is
believed to belong to the so-called Anisotropic KPZ (AKPZ) universality class
[Borodin and Ferrari, 2014]. It can be seen either as a two-dimensional
interacting particle process with drift, that generalizes the one-dimensional
Hammersley process [Aldous and Diaconis 1995, Seppalainen 1996], or as an
irreversible dynamics of lozenge tilings of the plane [Borodin and Ferrari
2014, Toninelli 2015]. Our main result is a hydrodynamic limit: the interface
height profile converges, after a hyperbolic scaling of space and time, to the
solution of a non-linear first order PDE of Hamilton-Jacobi type with
non-convex Hamiltonian (non-convexity of the Hamiltonian is a distinguishing
feature of the AKPZ class). We prove the result in two situations: (i) for
smooth initial profiles and times smaller than the time $T_{shock}$ when
singularities (shocks) appear or (ii) for all times, including $t>T_{shock}$,
if the initial profile is convex. In the latter case, the height profile
converges to the viscosity solution of the PDE. As an important ingredient, we
introduce a Harris-type graphical construction for the process.
