This uses a descending numerical sort and grabs the first element to get the largest integer. It also keeps the associated text that goes before the integer.

my $rxPair = qr{(?x) ( [A-Z]+ \s+ \d+ ) };

print
   map { qq{$_->[0]\n     @{ $_->[1] }\n} }
   map {
           my $raLine = $_;
           my $raMax =
              ( sort {$b->[1] <=> $a->[1] } @{ $raLine->[1] } )[0];
           [ $raLine->[0], $raMax ]
       }
   map { chomp; [ $_, [ map { [ split ] } m{$rxPair}g ] ] }
   <DATA>;

__END__
ASBSDEC 34 GADVVEEVEETTE 56 IOEOREAK GKJEOG EFEAF 1090 DAFFEE 376
ASB C 134 PPKOREAK EFEAF 290
BLAH 99 BLAH 123 FRED 27 BARNEY 427
[download]

Here's the output

ASBSDEC 34 GADVVEEVEETTE 56 IOEOREAK GKJEOG EFEAF 1090 DAFFEE 376
     EFEAF 1090
ASB C 134 PPKOREAK EFEAF 290
     EFEAF 290
BLAH 99 BLAH 123 FRED 27 BARNEY 427
     BARNEY 427
[download]

A little late but never mind.

Cheers,

JohnGG