Okay, I think I understand your requirements better.

How about the following ...?

my $MAL = '"MAL_[^"]*"';
while ( <LOG> ) {
  while ( $_ =~ /$MAL/gi ) {
    print OUT $_;
  }
}
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Now the regex is matching "MAL_ anywhere in the string, and ending with the matching quotation mark ", but containing no quotes in between.  The regular expression substring [^"] means "a character which isn't a quotation mark", and [^"]* means "any number of non-quotation mark characters".

Is that what you want?