The charmless baryonic decay $B^+\to \bar{\Lambda} p\bar{p}p$ is observed using proton-proton collision data recorded by the LHCb experiment, corresponding to an integrated luminosity of 5.4 $\text{fb}^{-1}$. The branching fraction for this decay is measured for the first time relative to that of the topologically similar decay $B^+\to J/\psi K^+$, with $J/\psi \to \bar \Lambda p K^-$. The background-subtracted invariant-mass distributions of $\bar{\Lambda} p $ and $\bar p p $ pairs exhibit pronounced enhancements at both kinematic thresholds, in contrast to a uniform phase-space distribution. A measurement of the $CP$ asymmetry for this decay is also reported.