Transition points mark qualitative changes in the macroscopic properties of large complex systems. Explosive transitions, exhibiting properties of both continuous and discontinuous phase transitions, have recently been uncovered in network growth processes. Real networks not only grow but often also restructure; yet common network restructuring processes, such as small world rewiring, do not exhibit phase transitions. Here, we uncover a class of intrinsically discontinuous transitions emerging in network restructuring processes controlled by adhesion - the preference of a chosen link to remain connected to its end node. Deriving a master equation for the temporal network evolution and working out an analytic solution, we identify genuinely discontinuous transitions in nongrowing networks, separating qualitatively distinct phases with monotonic and with peaked degree distributions. Intriguingly, our analysis of empirical data indicates a separation between the same two forms of degree distributions distinguishing abstract from face-to-face social networks.