This morning I ran
sudo apt-get upgrade and it prompted me to update
gnome-shell. I never really use the gnome desktop environment, but I figured I might as well keep it up-to-date.
After restarting my laptop I noticed that lightdm had selected gnome as the desktop environment and when I tried to switch it back to unity I noticed that unity was missing from the list.
Unity is still installed somewhere, but it seems to have been removed from
/usr/share/xsessions/. I've tried running
sudo apt-get install --reinstall unity, but it doesn't seem to have changed anything.
I've checked this answer here and it suggests that the solution is to remove gnome-shell. I'm a bit hesitant to do this though, if I remove gnome-shell and unity doesn't come back I'll be left without any desktop environment.