  In recent times, the theoretical study of the three-dimensional
Edwards-Anderson model has produced several rigorous results on the nature of
the spin-glass phase. In particular, it has been shown that, as soon as the
overlap distribution is non-trivial, ultrametricity holds. However, these
theorems are valid only in the thermodynamical limit and are therefore of
uncertain applicability for (perennially off-equilibrium) experimental spin
glasses. In addition, their basic assumption of non-triviality is still hotly
debated. This paper intends to show that the predictions stemming from
ultrametricity are already well satisfied for the lattice sizes where numerical
simulations are possible (i.e., up to $V = 32^3$ spins) and are, therefore,
relevant at experimental scales. To this end we introduce a three-replica
correlation function, which evinces the ultrametric properties of the system
and is shown to scale in the same way as the overlap correlation function.
