Update: Greatly simplified the else case.

Try this:

#! perl -slw
use strict;
use Data::Dump qw[ pp ];
use List::Util qw[ sum ];

sub rankSums {
    my( $aRef, $bRef ) = @_;
    my( $aSum, $bSum ) = (0) x 2;
    my( $a, $b ) = (0) x 2;
    my $rank = 1;
    while( $a < @$aRef && $b < @$bRef ) {
        if( $aRef->[ $a ] < $bRef->[ $b ] ) {
            $aSum += $rank++;
            ++$a;
        }
        elsif( $aRef->[ $a ] > $bRef->[ $b ] ) {
            $bSum += $rank++;
            ++$b
        }
        else {
            $aSum += ( $rank * 2 + 1 ) / 2;
            $bSum += ( $rank * 2 + 1 ) / 2;
            $rank += 2;
            ++$a, ++$b;
       }
    }
    $aSum += $rank++ while $a++ < @{ $aRef };
    $bSum += $rank++ while $b++ < @{ $bRef };

    return $aSum, $bSum;
}

my @a = split ' ', <DATA>;
my @b = split ' ', <DATA>;

my( $aSum, $bSum ) = rankSums( \@a, \@b );

print "asum:$aSum bSum:$bSum";


#__DATA__
#1 3 5 7 9
#2 4 6 8 10

#__DATA__
#1 2 3 4 5 6 7 8 9 10
#3.14 4.25 5.36 6.47 7.58

__DATA__
1 2 3 4 5
3 3.14 4 4
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It makes a single pass over the data, and does no copying or sorting or memory allocation, so it should be considerably faster than the current method, but full testing & benchmarking it is your task :)

A further simplified version of the function that runs a tad quicker and has been tested with 1000 runs of 1e6 x 1e6 random integers:

sub rankSums3 {
    my( $aRef, $bRef ) = @_;
    my( $aSum, $bSum ) = (0) x 2;
    my( $a, $b ) = (0) x 2;
    my $rank = 1;
    while( $a < @$aRef && $b < @$bRef ) {
        $aSum += $rank++, ++$a, next if $aRef->[ $a ] < $bRef->[ $b ];
        $bSum += $rank++, ++$b, next if $aRef->[ $a ] > $bRef->[ $b ];
        $aSum += ( $rank * 2 + 1 ) / 2;
        $bSum += ( $rank * 2 + 1 ) / 2;
        $rank += 2;
        ++$a, ++$b;
    }
    $aSum += $rank++ while $a++ < @{ $aRef };
    $bSum += $rank++ while $b++ < @{ $bRef };

    return $aSum, $bSum;
}
[download]