This works for me:

use IO::Handle;

my $file = 'output.dat';

open(OUTPUT, ">$file")   || die $!;
STDOUT->fdopen(\*OUTPUT, "w")  || die $!;

open STDERR, ">&STDOUT";

print        "This is STDOUT\n";
print STDERR "This is STDERR\n";
warn         "This is a warning\n";
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Update: Cleaner:

my $file = 'output.dat';

{
   open(my $fh_out, '>', $file)
      or die("Unable to create output file \"$file\": $!\n");
 
   local *STDOUT = $fh_out;
   local *STDERR = $fh_out;

   print        "This is STDOUT (1)\n";
   print STDERR "This is STDERR (1)\n";
   warn         "This is a warning (1)\n";
}

print        "This is STDOUT (2)\n";
print STDERR "This is STDERR (2)\n";
warn         "This is a warning (2)\n";
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If you want child processes to inherit STDOUT and STDERR:

use v5.8.0;

my $file = 'output.dat';

{
   open(my $fh_out, '>', $file)
      or die("Unable to create output file \"$file\": $!\n");
 
   local *STDOUT; open(*STDOUT, '>&', $fh_out)
      or die("Unable to redirect STDOUT: $!\n");

   local *STDERR; open(*STDERR, '>&', $fh_out)
      or die("Unable to redirect STDERR: $!\n");

   print        "This is STDOUT (1)\n";
   print STDERR "This is STDERR (1)\n";
   warn         "This is a warning (1)\n";
}

print        "This is STDOUT (2)\n";
print STDERR "This is STDERR (2)\n";
warn         "This is a warning (2)\n";
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