  We study the conventional holographic recipes and its real-time extensions in
the context of the correspondence between SYK quantum mechanics and JT gravity.
We first observe that only closed contours are allowed to have a 2d space-time
holographic dual. Thus, in any real-time formulation of the duality, the
boundaries of a classical connected geometry are a set of closed curves,
parameterized by a complex \emph{closed} time contour as in the
Schwinger-Keldysh framework. Thereby, a consistent extension of the standard
holographic formulas is proposed, in order to describe the correspondence
between gravity and boundary quantum models that include averaging on the
coupling constants. We investigate our prescription in different AdS$_{1+1}$
solutions with Schwinger-Keldysh boundary condition, dual to a boundary quantum
theory at finite temperature defined on a complex time contour, and consider
also classical, asymptotically AdS solutions (wormholes) with two disconnected
boundaries. In doing this, we revisit the so-called factorization problem, and
its resolution in conventional holography by virtue of some (non-local)
coupling between disconnected boundaries, and we show how in specific contexts,
the averaging proposal by-passes the paradox as well, since it induces a
similar effective coupling.
