So here's what needs to happen (Dan and me debugged this far):
1. libsoup establishes a connection to HOST
2. The network dies
3. libsoup tries to establish another connection to HOST, which fails.
After this happens, libsoup tries repeatedly to create a new connection. In fact, it gets into a loop that makes Epiphany/Webkit not receive repaint or user input events anymore.

This is reproducible by loading Planet GNOME and at the right point during  loading, right-click => uncheck "Enable Networking" in nm-applet.

It's also happening in the real world quite common when suspending the laptop and resuming it someplace else without an active network connection.