  In linear perturbation theory, a static perturber in the vicinity of a
Schwarzschild black hole (BH) enhances [suppresses] the Gauss-Bonnet (GB)
curvature invariant, $\mathcal{R}_{\rm GB}$, in the high [low] tide regions. By
analysing exact solutions of the vacuum Einstein field equations describing one
or two BHs immersed in a multipolar gravitational field, which is locally free
of pathologies, including conical singularities, we study the corresponding
non-linear tides on a fiducial BH, in full General Relativity (GR). We show
that the tidal field due to a far away, or close by, static BH creates high/low
tides that can deviate not only quantitatively but also qualitatively from the
weak field/Newtonian pattern. Remarkably, the suppression in low tide regions
never makes $\mathcal{R}_{\rm GB}$ negative on the BH, even though the horizon
Gaussian curvature may become negative; but $\mathcal{R}_{\rm GB}$ can vanish
in a measure zero set, a feature qualitatively recovered in a Newtonian
analogue model. Thus, purely gravitational, static, tidal interactions in GR,
no matter how strong, cannot induce GB$^-$ scalarization. We also show that a
close by BH produces noticeable asymmetric tides on another (fiducial) BH.
