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 a π 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 π-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 fermion materials. The Ginzburg-Landau functional reveals not only that the π-triplet gap amplitude couples trilinearly with the singlet gap amplitude and the staggered magnetization magnitude but also that the π-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 antiparallel 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 π-triplet order parameter the coexistence region is enhanced. This effect stems from the fact that the π-triplet degrees of freedom promote an effective attraction between the antiferromagnetic and singlet superconducting degrees of freedom, highlighting the complex interplay between these two orders, which goes beyond mere competition for the same electronic states.