Open new terminal tab in the same directory with the active tab

Opening a new tab in Terminal should by default retain the current directory. If it doesn't, you may've broken the $PROMPT_COMMAND.

Also remember to not replace the previous value if you intend to add custom behaviour by adding ; $PROMPT_COMMAND at the end.

PROMPT_COMMAND="my_custom_function; $PROMPT_COMMAND"

For reference, here's the default from /etc/bashrc (OS X 10.9):

# Tell the terminal about the working directory at each prompt.
if [ "$TERM_PROGRAM" == "Apple_Terminal" ] && [ -z "$INSIDE_EMACS" ]; then
    update_terminal_cwd() {
        # Identify the directory using a "file:" scheme URL,
        # including the host name to disambiguate local vs.
        # remote connections. Percent-escape spaces.
        local SEARCH=' '
        local REPLACE='%20'
        local PWD_URL="file://$HOSTNAME${PWD//$SEARCH/$REPLACE}"
        printf '\e]7;%s\a' "$PWD_URL"
    }
    PROMPT_COMMAND="update_terminal_cwd; $PROMPT_COMMAND"
fi

So I found the problem with the Terminal and I'm posting the answer hoping it can help others. The problem was in the definition of the PROMPT_COMMAND environmental variable. I changed its definition from

export PROMPT_COMMAND='history -n;'

to

export PROMPT_COMMAND="${PROMPT_COMMAND:+$PROMPT_COMMAND ;} history -n"

appending at the end of the existing PROMPT_COMMAND instead and this solved the problem.