We study the possibility of producing a new kind of nuclear systems which in addition to ordinary nucleons contain a few antibaryons (B = p, Λ, etc.). The properties of such systems are described within the relativistic mean-field model by employing G-parity transformed interactions for antibaryons. Calculations are first done for infinite systems and then for finite nuclei from 4 He to 208 Pb. It is demonstrated that the presence of a real antibaryon leads to a strong rearrangement of a target nucleus resulting in a significant increase of its binding energy and local compression. Noticeable effects remain even after the antibaryon coupling constants are reduced by factor 3 − 4 compared to G-parity motivated values. We have performed detailed calculations of the antibaryon annihilation rates in the nuclear environment by applying a kinetic approach. It is shown that due to significant reduction of the reaction Q-values, the in-medium annihilation rates should be strongly suppressed leading to relatively long-lived antibaryonnucleus systems. Multi-nucleon annihilation channels are analyzed too. We have also estimated formation probabilities of bound B + A systems in pA reactions and have found that their observation will be feasible at the future GSI antiproton facility. Several observable signatures are proposed. The possibility of producing multi-quark-antiquark clusters is discussed.