You could probably do it with a series of settings, such as (and I haven't fully tested this)...
export PROMPT_COMMAND='history -a; history -c; history -r'
That will cause the most recent command to be appended to your
.bash_history file, then clear your shell history, then re-read the
.bash_history file, each time the shell prompt is printed... so each time you run a command.
So, commands from one terminal will be appended to
.bash_history every time they're run, then the second terminal will pick them up when it re-reads the history file.
Now, keep in mind that this is not realtime, as your second shell will not re-read the history file that was just written by the first shell until a command is run, but it's probably as close as you'll get with bash.
You may find value in setting your
HISTCONTROL envvar to
erasedups, or this option may not work so well for you.