The flip-flop operator returns a special value when the flop condition is true. Take advantage of it to determine that it's time to process what you have accumulated.

my $found;
while (<>) {
    my $i = /ABCD/ ... /EFGH/;
    if ($i) {
        $found .= $_;
        if ($i =~ /E/) {
            print "========\n";
            print $found;
            $found = '';
        }
    }
}
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You can also achieve the same with the following:

while (<>) {
    my $i = /ABCD/ ... /EFGH/;
    print "========\n" if $i == 1;
    print if $i;
}
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