We adopt a geometric perspective on Fock space to provide two complementary insights into the eigenstates in many-body localized fermionic systems. On the one hand, individual many-body-localized eigenstates are well approximated by a Slater determinant of single-particle orbitals. On the other hand, the orbitals of different eigenstates in a given system display a varying, and generally imperfect, degree of compatibility, as we quantify by a measure based on the projectors onto the corresponding single-particle subspaces. We study this incompatibility between states of fixed and differing particle number, as well as inside and outside the many-body localized regime. This gives detailed insights into the emergence and strongly correlated nature of quasiparticlelike excitations in many-body localized systems, revealing intricate correlations between states of different particle numbers down to the level of individual realizations.