Source file in bash to set prompt with ANSI colors

You use single quotes in the PS1= assignments, preventing variable expansion from happening right there at assignment time. For PS1 that's fine as Bash will do a second expansion pass later, immediately before it prints the prompt.

But that second pass will happen after the function ends (the PS1 isn't printed by or during the PROMPT_COMMAND; it is printed after), and because you've declared all those variables local to the function, they cease to exist when the function ends – so when they're finally needed they all expand to empty values (or wrong values in case of $?).

Anything you use from within PS1 with the delayed expansion needs to be a global variable. So you need to un-local practically everything in that function, and replace direct usage of $? with the stored $EXIT.

# These are constant so they can be declared once, outside the function
# If you want you can even 'readonly RED=...' to truly make them constant
CLEAR='\[\e[0m\]'
RED='\[\e[1;31m\]'

__prompt_command() {
    # Without 'local' the assignment will be global by default
    # You can use 'declare -g' to explicitly say "this is *deliberately* global”
    EXIT="$?"

In some cases I'd even use a function from within PS1, i.e. PS1='$(_print_cool_prompt)' and let that function directly 'echo' or 'printf' the prompt. (In such cases it would need to print \001..\002 around the color codes, not the usual \[..\].)

Also, you have too many backslashes in some of your color codes. The \[...\] are okay as they have special meaning around color codes. But that means there shouldn't be a \[ within the color code. It needs to be just \e[, not \e\[.

Also, try \e[91m for bright red that isn't bold – 1;31 means "bold + red", and in many terminals bold also makes the color bright, but most terminals support color codes 90+ to get bright colors without necessarily making the text bold.

(Also, the usual style convention is that local variables are lower-case. I'd say even global variables, when they aren't exported into the environment, seem to be better as lower-case.)

@grawity's answer worked great! Thought I'd share the result and the code that made it:

__prompt_command() {
    EXIT="$?"
    CLEAR="\[\e[0m\]"
    RED="\[\e[91m\]"
    GREEN="\[\e[32m\]"
    CYAN="\[\e[33m\]"
    BLUE="\[\e[34m\]"
    MAGENTA="\[\e[35m\]"
    # bold
    REDB="\[\e[1;91m\]"
    GREENB="\[\e[1;32m\]"
    CYANB="\[\e[1;33m\]"
    BLUEB="\[\e[1;34m\]"
    MAGENTAB="\[\e[1;35m\]"

    local BGWHITE="\[\e[47m\]"

    GIT_BRANCH="$(git branch --show-current 2>/dev/null)"
    if [ "$GIT_BRANCH" == "" ]; then
        GIT_BRANCH=" "
    else
        GIT_BRANCH=" (${GIT_BRANCH}) "
    fi
    PS1="${BGWHITE}\t${CLEAR} ${CYAN}\w${CLEAR} ${MAGENTA}$GIT_BRANCH${CLEAR} ${RED}\u@\h${GREEN}\n\$${CLEAR} "

    if [ $EXIT != 0 ]; then
        PS1="[ ${GREENB}${EXIT}${CLEAR} ] ${PS1}"
    else
        PS1="[ ${REDB}${EXIT}${CLEAR} ] ${PS1}"
    fi
}
PROMPT_COMMAND=__prompt_command