We present joint NuSTAR and XMM-Newton observations of the bright, variable quasar IRAS 13349+2438. This combined dataset shows two clear iron absorption lines at 8 and 9 keV, which are most likely associated with two layers of mildly relativistic blueshifted absorption, with velocities of 0.14c and 0.27c. We also find strong evidence for a series of Ly$alpha$ absorption lines at intermediate energies in a stacked XMM-Newton EPIC-pn spectrum, at the same blueshift as the lower velocity iron feature. This is consistent with a scenario where an outflowing wind is radially stratified, so faster, higher ionization material is observed closer to the black hole, and cooler, slower material is seen from streamlines at larger radii.