  Motivated by the recent progress in developing high-precision relativistic
chiral nucleon-nucleon interactions, we study the antinucleon-nucleon
interaction at the leading order in the covariant chiral effective field
theory. The phase shifts and inelasticities with $J\leq 1$ are obtained and
compared to their non-relativistic counterparts. For most partial waves, the
descriptions of phase shifts and inelasticities in the leading-order covariant
chiral effective field theory are comparable to those in the next-to-leading
order non-relativistic chiral effective field theory, confirming the relatively
faster convergence observed in the nucleon-nucleon sector. In addition, we
search for bound states/resonances near the $\bar{N}N$ threshold and find
several structures that can be associated with those states recently observed
by the BESIII Collaboration.
