Taking a guess about your contradictory requirements:

#!/usr/bin/perl

# http://perlmonks.org/?node_id=1194583

use strict;
use warnings;

while(<DATA>)
  {
  s/ \b (?![A-Z]{2,}\b) (\w+) \b /\L$1/gx;
  print;
  }

__DATA__
I am a GIRL
A test WITH Multiple conDITions AND other tests.
[download]

use warnings;
use strict;

my $text = "I am a GIRL, a Real GIRL. A test WITH Multiple conDITions 
+AND other tests.";

print map{"$_ "}
      map{$_ = lc($_) unless (($_ eq uc $_) and length $_ > 1 ); $_ }
      map{split ' ',$_}$text;
      
__END__
Prints: 
i am a GIRL, a real GIRL. a test WITH multiple conditions AND other te
+sts.
[download]

Update: Here is one possibility of many:

use warnings;
use strict;

my $text = "I am a GIRL, a Real GIRL. A test WITH Multiple conDITions 
+AND other tests.";

print "".onlyAllcaps ($text);


sub onlyAllcaps
{
   my $input = shift;
   
   my @words = split ' ',$text;
   foreach my $word (@words)
   {
      $word = lc($word) unless (($word eq uc $word) and length $word> 
+1 )
   }

   my $output = join " ",@words,"\n";
   return $output;
}

__END__
Prints:
i am a GIRL, a real GIRL. a test WITH multiple conditions AND other te
+sts.
[download]

This is one of those cases where for readability sake, I would separate the split logic from the case logic. This isn't going to win any perl golf competitions but future maintainers will not try to hunt you down!

#!/usr/bin/env perl

use strict;
use warnings;

my $input = "I am a GIRL";
my $output = [];

# split input by non-words ... if you need to keep non-word 
# characters then you'll need a different regex
foreach my $word ( split( /\W+/, $input ) ) {
  # if the word is all upper case and it's not a single letter
  if( $word =~ /^\p{Uppercase}+$/ && length( $word ) > 1 ) {
    push( @$output, $word );
  } else {
    push( @$output, lc $word );
  }
}

print join( ' ', @$output ), "\n";
[download]

update: Fix regex from /\p{Lu}+/ to /^\p{Uppercase}+$/