  We find models of two-dimensional gravity that resolve the factorization
puzzle and have a discrete spectrum, whilst retaining a semiclassical
description. A novelty of these models is that they contain non-trivially
correlated spacetime branes or, equivalently, nonlocal interactions in their
action. Such nonlocal correlations are motivated in the low-energy gravity
theory by integrating out UV degrees of freedom. Demanding factorization fixes
almost all brane correlators, and the exact geometric expansion of the
partition function collapses to only two terms: the black hole saddle and a
subleading ``half-wormhole'' geometry, whose sum yields the desired discrete
spectrum. By mapping the insertion of correlated branes to a certain
double-trace deformation in the dual matrix integral, we show that
factorization and discreteness also persist non-perturbatively. While in our
model all wormholes completely cancel, they are still computationally relevant:
self-averaging quantities, like the Page curve, computed in the original theory
with wormholes, accurately approximate observables in our theory, which
accounts for UV corrections. Our models emphasize the importance of
correlations between different disconnected components of spacetime, providing
a possible resolution to the factorization puzzle in any number of dimensions.
