See the Tie::File documentation on recsep:

By default, the meaning is the same as for the <...> operator: It's a string terminated by $/, which is probably "\n". (Minor exception: on DOS and Win32 systems, a 'record' is a string terminated by "\r\n".)

So that's exactly your problem. Change recsep to \n and likely everything will "just work".

Change recsep to \n

I hope this won't be necessary, because I don't know in advance which line ending style the file will have. There is, however, something I don't quite understand: As you correctly say, the documentation explains By default, the meaning is the same as for the <...> operator. But now look at the following program, which uses the angle operator insead of tie, to read the same file:

use strict;
use warnings;

sub printfirst {
  my $f=shift;
  open(F,$f) or die "$!";
  my $firstline=<F>;
  print "$f: $firstline\n";
}
printfirst('ALSF.pm');
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In this case, $firstline correctly gets only the first line in the file, although the lines in the file are terminated by \n and we are running on Windows. Shouldn't then $firstline also contain the whole file in this case?
-- 
Ronald Fischer <ynnor@mm.st>

I guess that's some inexactness in the documentation of Tie::File. Looking at the source code, there is the following code, which sets/forces $/ to \r\n:

sub _default_recsep {
  my $recsep = $/;
  if ($^O eq 'MSWin32') {       # Dos too?
    # Windows users expect files to be terminated with \r\n
    # But $/ is set to \n instead
    # Note that this also transforms \n\n into \r\n\r\n.
    # That is a feature.
    $recsep =~ s/\n/\r\n/g;
  }
  $recsep;
}
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So, you'll have to read the first few bytes (or up to the first \n), guess the line ending from that, and then use the recsep option to force the line ending.