With my config a normal highlight interacts with tmux and if I hold shift it interacts with my terminal emulator. I'm using Termite, but I swear it worked the same in others like terminator and urxvt as well. I have both a local and remote tmux session and am basically exclusively using shells in tmux, so I've long since gotten used to in. This must be fairly standard, because a friend of mine uses Hexchat on Windows where a normal highlight can grab a line from IRC, but without the timestamp. Holding shift lets you grab all the actual text, timestamp included.