We analyze the eigenstates' Fock space structure in the tilted Bose-Hubbard Hamiltonian. Fock space localization is quantified in terms of finite-size generalized fractal dimensions. This approach allows us to monitor the parametric evolution of localization across avoided crossings in the energy spectrum. We identify a manifold of strongly localized eigenstates, both in Fock and configuration space, which persists in regimes where all system parameters (hopping, interaction, and tilt strengths) are comparable. For a fixed number of bosons, the localization properties of that manifold are independent of the lattice size. The influence of avoided crossings on the structure of the localized manifold is discussed in detail.