  In this paper, we consider vacuum asymptotically anti-de Sitter spacetimes $(
\mathscr{M}, g )$ with conformal boundary $( \mathscr{I}, \mathfrak{g} )$. We
establish a correspondence, near $\mathscr{I}$, between such spacetimes and
their conformal boundary data on $\mathscr{I}$. More specifically, given a
domain $\mathscr{D} \subset \mathscr{I}$, we prove that the coefficients
$\mathfrak{g}^{(0)} = \mathfrak{g}$ and $\mathfrak{g}^{(n)}$ (the undetermined
term or stress energy tensor) in a Fefferman-Graham expansion of the metric $g$
from the boundary uniquely determine $g$ near $\mathscr{D}$, provided
$\mathscr{D}$ satisfies a generalised null convexity condition (GNCC). The GNCC
is a conformally invariant criterion on $\mathscr{D}$, first identified by
Chatzikaleas and the second author, that ensures a foliation of pseudoconvex
hypersurfaces in $\mathscr{M}$ near $\mathscr{D}$, and with the pseudoconvexity
degenerating in the limit at $\mathscr{D}$. As a corollary of this result, we
deduce that conformal symmetries of $( \mathfrak{g}^{(0)}, \mathfrak{g}^{(n)}
)$ on domains $\mathscr{D} \subset \mathscr{I}$ satisfying the GNCC extend to
spacetimes symmetries near $\mathscr{D}$. The proof, which does not require any
analyticity assumptions, relies on three key ingredients: (1) a calculus of
vertical tensor-fields developed for this setting; (2) a novel system of
transport and wave equations for differences of metric and curvature
quantities; and (3) recently established Carleman estimates for tensorial wave
equations near the conformal boundary.
