Urm... the filehandle for stat, actually for -s, works for me.

open OUT, ">test.txt";
for (1 .. 1000) {
    print OUT "Hello, Perlmonks!\n" x 10;
    print -s OUT;
}
[download]

It works, but -s is indeed set according to the buffer size, and not according to the already printed output:

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4096
4096
4096
4096
4096
4096
4096
4096
...

With $| set to a true value it'll work less coarse, but most likely quite a bit slower:

open OUT, ">test.txt";
my $fh = select OUT;
$| = 1;
select $fh;
for (1 .. 1000) {
    print OUT "Hello, Perlmonks!\n" x 10;
    print -s OUT;
}
[download]

192
384
576
768
960
1152
1344
1536
1728
1920
2112
2304
...

I think the coarse version is still fine enough.