Is there a way to share the shell command history between panes/windows in a tmux session?
shell history has precious little to do with tmux, it has to do with the shell you are using. So if you chose to use zsh it is enabled iirc by default. With bash you need to add some magic to your .bashrc
export PROMPT_COMMAND="history -a; history -n"
this appends your last command to history and reloads your history after each command. See this post for more information.