Silicon carbide (SiC) is an excellent substrate for growth and manipulation of large scale, high quality epitaxial graphene. On the carbon face (the ($bar{1}bar{1}bar{1}$) or $(000bar{1}$) face, depending on the polytype), the onset of graphene growth is intertwined with the formation of several competing surface phases, among them a (3$times$3) precursor phase suspected to hinder the onset of controlled, near-equilibrium growth of graphene. Despite more than two decades of research, the precise atomic structure of this phase is still unclear. We present a new model of the (3$times$3)-SiC-($bar{1}bar{1}bar{1}$) reconstruction, derived from an {it ab initio} random structure search based on density functional theory including van der Waals effects. The structure consists of a simple pattern of five Si adatoms in bridging and on-top positions on an underlying, C-terminated substrate layer, leaving one C atom per (3$times$3) unit cell formally unsaturated. Simulated scanning tunneling microscopy (STM) images are in excellent agreement with previously reported experimental STM images.
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