  We thoroughly examine the ground state of the triangular lattice of Pb on
Si(111) using scanning tunneling microscopy. We detect charge-order,
accompanied by a subtle structural reconstruction. Applying the extended
variational cluster approach we map out the phase diagram as a function of
local and non-local Coulomb interactions. Comparing the experimental data with
the theoretical modeling leads us to conclude that electron correlations are
the driving force of the charge-ordered state in Pb/Si(111), rather than Fermi
surface nesting. These results resolve the discussion about the origin of the
well known $3\times 3$ reconstruction forming below $86\,$K. By exploiting the
tunability of correlation strength, hopping parameters and bandfilling, this
material class represents a promising platform to search for exotic states of
matter, in particular, for chiral topological superconductivity.
