  The close interplay between superconductivity and antiferromagnetism in
several quantum materials can lead to the appearance of an unusual
thermodynamic state in which both orders coexist microscopically, despite their
competing nature. A hallmark of this coexistence state is the emergence of a
spin-triplet superconducting gap component, called $\pi$-triplet, which is
spatially modulated by the antiferromagnetic wave-vector, reminiscent of a
pair-density wave. In this paper, we investigate the impact of these
$\pi$-triplet degrees of freedom on the phase diagram of a system with
competing antiferromagnetic and superconducting orders. Although we focus on a
microscopic two-band model that has been widely employed in studies of iron
pnictides, most of our results follow from a Ginzburg-Landau analysis, and as
such should be applicable to other systems of interest, such as cuprates and
heavy fermions. The Ginzburg-Landau functional reveals not only that the
$\pi$-triplet gap amplitude couples tri-linearly with the singlet gap amplitude
and the staggered magnetization magnitude, but also that the $\pi$-triplet
$d$-vector couples linearly with the magnetization direction. While in the mean
field level this coupling forces the $d$-vector to align parallel or
anti-parallel to the magnetization, in the fluctuation regime it promotes two
additional collective modes - a Goldstone mode related to the precession of the
$d$-vector around the magnetization and a massive mode, related to the relative
angle between the two vectors, which is nearly degenerate with a Leggett-like
mode associated with the phase difference between the singlet and triplet gaps.
We also investigate the impact of magnetic fluctuations on the
superconducting-antiferromagnetic phase diagram, showing that due to their
coupling with the $\pi$-triplet order parameter, the coexistence region is
enhanced.
