@files = glob "*.xml";

undef $/;
for $file (@files) {
    $indent = 0;
    open FILE, $file or die "Couldn't open $file for reading: $!";
    $_ = readline *FILE;
    close FILE or die "Couldn't close $file: $!";

    # Remove whitespace between > and < if that is the only thing separating
    # them
    s/(?<=>)\s+(?=<)//g;

    # Indent
    s{  # Capture a tag <$1$2$3>,
        # a potential closing slash $1
        # the contents $2
        # a potential closing slash $3
        <(/?)([^/>]+)(/?)> 

        # Optional white space
        \s*

        # Optional tag.
        # $4 contains either undef, "<" or "</"
        (?=(</?))?
    }
    {
        # Adjust the indentation level.
        # $3: A <foo/> tag. No alteration to indentation.
        # $1: A closing </foo> tag. Drop one indentation level
        # else: An opening <foo> tag. Increase one indentation level
        $indent +=
            $3 ?  0 :
            $1 ? -1 :
                  1;

        # Put the captured tag back into place
        "<$1$2$3>" .
        # Two closing tags in a row. Add a newline and indent the next line
        ($1 and ($4 eq "</") ?
            "\n" . ("  " x $indent) : 
        # This isn't a closing tag but the next tag is. Add a newline and
        # indent the next line.
        $4 ?
            "\n" . ("  " x $indent) :
        # This isn't a closing tag - no special indentation. I forget why
        # this works.
            ""
        )
    # /g repeat as necessary
    # /e Execute the block of perl code to create replacement text
    # /x Allow whitespace and comments in the regex
    }gex;

    open FILE, ">", $file or die "Couldn't open $file for writing: $!";
    print FILE or die "Couldn't write to $file: $!";
    close FILE or die "Couldn't close $file: $!";
}
__END__

This is the version I copied and pasted from the working script.
Its ugly so I purtied it up and posted the version you see
above this text. I'm leaving this ugly version here just in case I
introduced some bug I'm not aware of.

@files = glob "*.xml";

undef $/;
$tag = "<>/";
for $file (@files) {
    $indent = 0;
    open FILE, $file;
    $_ = <FILE>;
    s/(?<=>)\s+(?=<)//g;
    s(<(/?)([^/>]+)(/?)>\s*(?=(</?))?)($indent+=$3?0:$1?-1:1;"<$1$2$3>".($1&&($4 eq"</")?"\n".("  "x$indent):$4?"\n".("  "x$indent):""))ge;
    open FILE, ">$file";
    print FILE;
}