Based on your suggestion, I developed the workaround below. The trick seems to have three parts to it:

1. binmode $FH, ':raw': binmode the tempfile(handle) before writing to it. (I suspect that on Unix, we can get away without the ':raw', but whatever.)

2. close $FH: close the tempfile(handle) after writing to it and before calling hexdigest on it.

3. Ignore File::Compare::compare() for now. (I don't need for my real-world problem, anyway.)

#!perl
use 5.14.0;
use warnings;
use Carp;
use Data::Dumper;
use Digest::MD5;
use File::Temp qw( tempfile );
use Test::More;

sub hexdigest_one_file {
    my $filename = shift;
    say "Filename: $filename";
    my $state = Digest::MD5->new();
    open my $FH, '<', $filename or croak "Unable to open $filename for
+ reading";
    $state->addfile($FH);
    close $FH or croak "Unable to close $filename after reading";
    return $state->hexdigest;
}

my $basic       = 'x' x 10**2;
my @digests;

my ($fh1, $t1) = tempfile();
binmode $fh1, ':raw';
for (1..100) { say $fh1 $basic }
close $fh1 or croak "Unable to close $t1 after writing";
push @digests, hexdigest_one_file($t1);

my $t3 = File::Temp->new( UNLINK => 0);
binmode $t3, ':raw';
for (1..100) { say $t3 $basic }
close $t3 or croak "Unable to close $t3 after writing";
push @digests, hexdigest_one_file($t3);

say Dumper [ @digests ];

cmp_ok($digests[0], 'eq', $digests[1],
    "Same md5_hex for $t1 and $t3");

done_testing();
[download]

Why this works I do not know. I think this is, at the very least, a deficiency in the File::Temp documentation and will file a bug report on it.

Thank you for your assistance.