F-injectivity does not imply F-fullness in normal domains

We construct examples of noetherian three-dimensional local geometrically normal domains of prime characteristic which are F-injective but not F-full. Along the way, we find examples of two-dimensional local geometrically normal domains which are F-injective but not F-anti-nilpotent. A crucial theme of our constructions is the behavior of F-injectivity along a purely inseparable finite base change.