Update: fixed code to work with zby's case and hopefully any other case.
Update: code won't work with duplicates in the second list

Not exactly vastly mystical but this should do the trick (although it hasn't been thoroughly tested)

use strict;

my @a = qw/ fred bob joe jim mary elaine /;
my @b = qw/ frank joe jim mary bob /;

print "LCS[anjiro] - ", join($", get_lcs(\@a, \@b)), $/;

@a = qw/ a b c /;
@b = qw/ a b x c /;

print "LCS[zby]    - ", join($", get_lcs(\@a, \@b)), $/;

sub get_lcs {
  my @a = @{ +shift };
  my @b = @{ +shift };

  my %map = map { $b[$_] => $_ } 0 .. $#b;

  my(@lcs, @tmp);
  for(0 .. $#a) {
    next unless exists $map{$a[$_]}
                or $a[$_ + 1] eq $b[$map{$a[$_]} + 1];

    push @tmp, $a[$_] if $a[$_] eq $b[$map{$a[$_]}];

    if($a[$_ + 1] ne $b[$map{$a[$_]} + 1]) {
      @lcs = @tmp if @tmp > @lcs;
      @tmp = ();
    }
  }
  @lcs = @tmp if @tmp > @lcs;

  return @lcs;
}

__output__

LCS[anjiro] - joe jim mary
LCS[zby]    - a b
[download]

You might also find some roughly applicable questions under Longest Common Substring.
HTH

_________
broquaint