  Results for the antinucleon-nucleon ($\bar NN$) interaction obtained at
next-to-next-to-next-to-leading order in chiral effective field theory (EFT)
are reported. A new local regularization scheme is used for the pion-exchange
contributions that has been recently suggested and applied in a pertinent study
of the $NN$ force within chiral EFT. Furthermore, an alternative strategy for
estimating the uncertainty is utilized that no longer depends on a variation of
the cutoffs. The low-energy constants associated with the arising contact terms
are fixed by a fit to the phase shifts and inelasticities provided by a
phase-shift analysis of $\bar pp$ scattering data. An excellent description of
the $\bar NN$ amplitudes is achieved at the highest order considered. Moreover,
because of the quantitative reproduction of partial waves up to $J=3$, there is
also a nice agreement on the level of $\bar pp$ observables. Specifically,
total and integrated elastic and charge-exchange cross sections agree well with
the results from the partial-wave analysis up to laboratory energies of $300$
MeV, while differential cross sections and analyzing powers are described
quantitatively up to $200$-$250$ MeV. The low-energy structure of the $\bar NN$
amplitudes is also considered and compared to data from antiprotonic hydrogen.
