  We derive higher-order corrections in the magnon dispersion relations for
two- and three-dimensional antiferromagnets exposed to magnetic and staggered
fields that are mutually aligned. "Dressing" the magnons is the prerequisite to
separate the low-temperature representation of the pressure into a piece due to
noninteracting magnons and a piece that corresponds to the magnon-magnon
interaction. Both in two and three spatial dimensions, the interaction in the
pressure turns out to be attractive. While concrete figures refer to the
spin-$\frac{1}{2}$ square-lattice and the spin-$\frac{1}{2}$ simple cubic
lattice antiferromagnet, our results are valid for arbitrary bipartite
geometry.
