  The direct long-term changes occurring in the orbital dynamics of a local
gravitationally bound binary system $S$ due to the post-Newtonian tidal
acceleration caused by an external massive source are investigated. A class of
systems made of a test particle $m$ rapidly orbiting with orbital frequency
$n_{\rm b}$ an astronomical body of mass $M$ which, in turn, slowly revolves
around a distant object of mass $M^{'}$ with orbital frequency $n_{\rm
b}^{'}\ll n_{\rm b}$ is considered. The characteristic frequencies of the
non-Keplerian orbital variations of $m$ and of $M$ itself are assumed to be
negligible with respect to both $n_{\rm b}$ and $n_{\rm b}^{'}$. General
expressions for the resulting Newtonian and post-Newtonian tidal orbital shifts
of $m$ are obtained. The future missions BepiColombo and JUICE to Mercury and
Ganymede, respectively, are considered in view of a possible detection. The
largest effects, of the order of $\approx 0.1-0.5$ milliarcseconds per year
(mas yr$^{-1}$), occur for the Ganymede orbiter of the JUICE mission. Although
future improvements in spacecraft tracking and orbit determination might,
perhaps, reach the required sensitivity, the systematic bias represented by the
other known orbital perturbations of both Newtonian and post-Newtonian origin
would be overwhelming. The realization of a dedicated artificial mini-planetary
system to be carried onboard and Earth-orbiting spacecraft is considered as
well. Post-Newtonian tidal precessions as large as $\approx 1-10^2$ mas
yr$^{-1}$ could be obtained, but the quite larger Newtonian tidal effects would
be a major source of systematic bias because of the present-day percent
uncertainty in the product of the Earth's mass times the Newtonian
gravitational parameter.
