The edge-product space of phylogenetic trees is a regular CW complex whose maximal closed cells correspond to trivalent trees with leaves labeled by a finite set $X$. The face poset of this cell decomposition is isomorphic to the Tuffley poset, a poset of labeled forests, with a unique minimum adjoined. We show that the edge-product space of phylogenetic trees is gallery-connected. We then use combinatorial properties of the Tuffley poset and a related graph known as NNI-tree space to show that, although open intervals of the Tuffley poset were proven to be shellable by Gill, Linusson, Moulton, and Steel, the edge-product space is not shellable.