use strict;
use warnings;

my $tries = 4;
my $sleep = 10;

my $req = HTTP::Request->new(GET => $reqstr);
$req->authorization_basic('username', 'password');

my $res;

my $attempts;
my $success;

for (;;) {
   ++$attempts;
   $res = $ua->request($req);

   if ($res->is_success) {
      my $title = $res->Header('title');

      if ($title ne 'Empty result') {
         $success = 1;
         last;
      }

#else { print("Error: Empty results\n"); }
   }

#else { print "HTTP Error: " . $res->status_line . "\n"; }

   last if $attempts >= $tries;

   # Sleep 10 seconds and try again.
#print("Trying again in $sleep seconds...\n");
   sleep($sleep);
}

die("Unable to download data after $tries tries.\n");
   unless $success;

open(my $out_fh, '>', $outfile)
or die("Could not open $outfile for writing: $!\n");

print $out_fh $req->content;
[download]

In your original code, you called $ua->request($req) 4 times before the while. That means that before you even started looping, you were requesting the page 4 times!!