While attractive, this is just one more way of tieing the STDERR glob.   Because this nor any of the other tie methods 'really' hook into the filehandle used by STDERR they don't help capture the low-level references to STDERR (aka fileno(STDERR)==2) done by code like Devel::Peek.

Tested with the following code it fails the "warn test" and testing versus Mr. Muskrat's target Devel::Peek:

use IO::Capture::Stderr;
use Devel::Peek;
    my  $data;
    print STDERR " STDERR test (1)\n";
    my  $capture = IO::Capture::Stderr->new();
    $capture->start();
    print STDERR " STDERR test (2)\n";
    warn " STDERR test (2a)";
    my $test = 'string';
    Dump($test);
    $capture->stop();
    print STDERR " STDERR test (3)\n";
    my  @the_output = $capture->read;
    if( @the_output ) {
        foreach (@the_output) {
            print "  >>> ", $_;
        }
    }
[download]

produces output

 STDERR test (1)
 STDERR test (2a) at mrmskrat4.pl line 15.
SV = PV(0x183ec0c) at 0x197138c
  REFCNT = 1
  FLAGS = (PADBUSY,PADMY,POK,pPOK)
  PV = 0x196deb4 "string"\0
  CUR = 6
  LEN = 7
 STDERR test (3)
  >>>  STDERR test (2)