  We study a general relativistic gravitomagnetic 3-body effect induced by the
spin angular momentum ${\boldsymbol S}_\textrm{X}$ of a rotating mass
$M_\textrm{X}$ orbited at distance $r_\textrm{X}$ by a local gravitationally
bound restricted two-body system $\mathcal{S}$ of size $r\ll r_\textrm{X}$
consisting of a test particle revolving around a massive body $M$. At the
lowest post-Newtonian order, we analytically work out the doubly averaged rates
of change of the Keplerian orbital elements of the test particle by finding
non-vanishing long-term effects for the inclination $I$, the node $\Omega$ and
the pericenter $\omega$. Such theoretical results are confirmed by a numerical
integration of the equations of motion for a fictitious 3-body system. We
numerically calculate the magnitudes of the post-Newtonian gravitomagnetic
3-body precessions for some astronomical scenarios in our solar system. For
putative man-made orbiters of the natural moons Enceladus and Europa in the
external fields of Saturn and Jupiter, the relativistic precessions due to the
angular momenta of the gaseous giant planets can be as large as $\simeq
10-50~\textrm{milliarcseconds~per~year}~\left(\textrm{mas~yr}^{-1}\right)$. A
preliminary numerical simulation shows that, for certain orbital configurations
of a hypothetical Europa orbiter, its range-rate signal $\Delta\dot\rho$ can
become larger than the current Doppler accuracy of the existing spacecraft Juno
at Jupiter, i.e. $\sigma_{\dot\rho}=0.015~\textrm{mm~s}^{-1}$, after 1 d. The
effects induced by the Sun's angular momentum on artificial probes of Mercury
and the Earth are at the level of $\simeq
1-0.1~\textrm{microarcseconds~per~year}~\left(\mu\textrm{as~yr}^{-1}\right)$.
