Between 2012 July and 2013 February, NuSTAR and XMM-Newton performed four long-look joint observations of the type 1.8 Seyfert, NGC 1365. We have analyzed the variable absorption seen in these observations in order to characterize the geometry of the absorbing material. Two of the observations caught NGC 1365 in an unusually low absorption state, revealing complexity in the multi-layer absorber that had previously been hidden. We find the need for three distinct zones of neutral absorption in addition to the two zones of ionized absorption and the Compton-thick torus previously seen in this source. The most prominent absorber is likely associated with broad-line region clouds with column densities of around ~1023 cm-2 and a highly clumpy nature as evidenced by an occultation event in 2013 February. We also find evidence of a patchy absorber with a variable column around ~1022 cm-2 and a line-of-sight covering fraction of 0.3-0.9, which responds directly to the intrinsic source flux, possibly due to a wind geometry. A full-covering, constant absorber with a low column density of ~1 × 1022 cm-2 is also present, though the location of this low density haze is unknown.