First off, there is not a clean solution to your problem without reimplementing some key component of how the shell (bash in this case) deals with history. Below is a solution that maintains your local history so that the arrows work as expected. Ctrl-r in turn is bound to searching your global history. The solution depends on an excellent utility for searching through your history called hh. Below are the instructions to set everything up.
Add to a startup file either
# Whenever a command is executed, write it to a global history
export PROMPT_COMMAND="history -a ~/.bash_history.global"
# On C-r run the swap_history_reverse.sh script,
bind -x '"\C-r": "~/swap_history_reverse.sh"'
Create the following script
#! /usr/bin/env bash
# Point hh to global history
# Reverse search
# Restore local history
Make the script executable:
chmod +x ~/swap_history_reverse.sh
hh, see INSTALL.
Once everything is in place, open up a new shell and give it a run.