  We calculate the effect of self-interaction on the "geodetic" spin precession
of a compact body in a strong-field orbit around a black hole. Specifically, we
consider the spin precession angle $\psi$ per radian of orbital revolution for
a particle carrying mass $\mu$ and spin $s \ll (G/c) \mu^2$ in a circular orbit
around a Schwarzschild black hole of mass $M \gg \mu$. We compute $\psi$
through $O(\mu/M)$ in perturbation theory, i.e, including the correction
$\delta\psi$ (obtained numerically) due to the torque exerted by the
conservative piece of the gravitational self-field. Comparison with a
post-Newtonian (PN) expression for $\delta\psi$, derived here through 3PN
order, shows good agreement but also reveals strong-field features which are
not captured by the latter approximation. Our results can inform
semi-analytical models of the strong-field dynamics in astrophysical binaries,
important for ongoing and future gravitational-wave searches.
