[en] We performed a first-principles study of the structural, vibrational, electronic, and magnetic properties of NaMnF3 under applied isotropic pressure. We found that NaMnF3 undergoes a reconstructive phase transition at 8 GPa from the Pnma distorted perovskite structure toward the Cmcm postperovskite structure. This is confirmed by a sudden change of the Mn-F-Mn bondings where the crystal goes from corner-shared octahedra in the Pnma phase to edge-shared octahedra in the Cmcm phase. The magnetic ordering also changes from a G-type antiferromagnetic ordering in the Pnma phase to a C-type antiferromagnetic ordering in the Cmcm phase. Interestingly, we found that the high-spin d-orbital filling is kept at the phase transition which has never been observed in the known magnetic postperovskite structures. We also found a highly noncollinear magnetic ordering in the Cmcm postperovskite phase that drives a large ferromagnetic canting of the spins. We discuss the validity of these results with respect to the U and J parameters of the GGA+U exchange-correlation functional used in our study and conclude that large-spin canting is a promising property of the postperovskite fluoride compounds.