This is quite ugly, but I have to go to work and this is just the first thing that popped into my head. It uses the local $/ like you described.

#!/usr/bin/perl -w

use strict;

local $/ = "\nField 1:  ";

my $data = <DATA>;
$data =~ s/Field 1:  $//;
chop($data);
print "[$data]\n";

while (<DATA>) {
    s/Field 1:  $//;
    $_ = "Field 1:  $_";
    chop;
    print "[$_]\n";
}

__DATA__
Field 1:  abc
Field 2:  asdfasdf
Field 2:  asdfasdfase
Field 2:  aaa
Field 3:  ss
Field 1:  def
Field 2:  abc123
Field 3:  blah
Field 1:  asdfa
[download]

Output is:

[Field 1:  abc
Field 2:  asdfasdf
Field 2:  asdfasdfase
Field 2:  aaa
Field 3:  ss]
[Field 1:  def
Field 2:  abc123
Field 3:  blah]
[Field 1:  asdf]
[download]