A small userscript that stops Basecamp's My Tasks tray from closing on stray clicks — most importantly, it lets you Ctrl-click / ⌘-click a task to open it in a new tab without the tray snapping shut. It also keeps the tray open when you click empty space or press Escape (close it deliberately with the tray's Close button or the T hotkey).
It only runs on app.basecamp.com and only touches the tasks tray.
You need an extension that runs userscripts: