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Context. Relativistic magnetic reconnection studies have so far focused on symmetric configurations, where the upstream plasma has identical properties on the two sides of the layer. Yet, just like nonrelativistic reconnection on the dayside of the Earth's magnetosphere, relativistic reconnection can also operate at the interface between highly asymmetric environments. The boundary layer between a relativistic jet and an accretion flow forming around a supermassive black hole can present asymmetric configurations in terms of plasma composition, bulk velocity, temperature, and magnetization. Aims. We conducted the first study of relativistic magnetic reconnection where the upstream plasma is composed of electron-positron pairs on one side, and electrons and ions on the other. We also investigated the impact of a relativistic symmetric shear flow applied along the reconnecting field lines. Methods. We simulated magnetic reconnection using 2D particle-in-cell simulations. The initial setup was adapted from a classic Harris layer without a guide field, modified to accommodate plasma-composition and shear asymmetries in the upstream medium. Results. For a composition-asymmetric setup, we find that the reconnection dynamics is driven by the electron-ion side, which is the plasma with the lowest magnetization. The energy partition favors accelerating ions at the expense of electrons even more than in a corresponding symmetric setup. With respect to shear, a super-Alfvénic upstream decreases the laboratory-frame reconnection rate, but, unlike in nonrelativistic studies, does not shut off reconnection completely. Conclusions. The asymmetries examined in this work lower the overall efficiency of electron acceleration relative to corresponding symmetric configurations. In the context of a black hole jet-disk boundary, asymmetric reconnection alone is probably not efficient at accelerating electrons to very high energies, but it might facilitate plasma mixing and particle injection for other acceleration channels at the interface. [ABSTRACT FROM AUTHOR]