  This article generalizes the conditional probability interpretation of time
in which time evolution is realized through entanglement between a clock and a
system of interest. This formalism is based upon conditioning a solution to the
Wheeler-DeWitt equation on a subsystem of the Universe, serving as a clock,
being in a state corresponding to a time $t$. Doing so assigns a conditional
state to the rest of the Universe $|\psi_S(t)\rangle$, referred to as the
system. We demonstrate that when the total Hamiltonian appearing in the
Wheeler-DeWitt equation contains an interaction term coupling the clock and
system, the conditional state $|\psi_S(t)\rangle$ satisfies a time-nonlocal
Schr\"{o}dinger equation in which the system Hamiltonian is replaced with a
self-adjoint integral operator. This time-nonlocal Schr\"{o}dinger equation is
solved perturbatively and three examples of clock-system interactions are
examined. One example considered supposes that the clock and system interact
via Newtonian gravity, which leads to the system's Hamiltonian developing
corrections on the order of $G/c^4$ and inversely proportional to the distance
between the clock and system.
