In theory you could write a TAP::Formatter::Base subclass, but it seems easier to copy and paste the code from Test::More::_deep_check() and modify it to suit your needs — changing how differences are reported.

use v5.12.0;
use warnings;
use Data::Dump qw/pp dd/;

use Test::More  import => ['is_deeply', 'done_testing'];

my $test_builder = Test::More->builder;

my %out;

$test_builder->output(\ $out{std});
$test_builder->failure_output(\ $out{failure});
$test_builder->todo_output(\ $out{todo});


warn "Test pass: " . is_deeply( {a=>1}, {a=>1},"pass?");
warn "Test fail: " . is_deeply( {a=>1}, {b=>1},"fail?");

done_testing;

my $div = "-"x10;

for my $type (qw/std failure todo/){
    
    my $out = $out{$type} // "";
    say <<"__OUT__";
$div $type $div
$out
$div
__OUT__
}

pp \%out;
[download]

Test pass: 1 at d:/temp/redirect_test_more.pl line 16.
Test fail: 0 at d:/temp/redirect_test_more.pl line 17.
---------- std ----------
ok 1 - pass?
not ok 2 - fail?
1..2

----------

---------- failure ----------
#   Failed test 'fail?'
#   at d:/temp/redirect_test_more.pl line 17.
#     Structures begin differing at:
#          $got->{b} = Does not exist
#     $expected->{b} = '1'

----------

---------- todo ----------

----------

{
  failure => "#   Failed test 'fail?'\n#   at d:/temp/redirect_test_mo
+re.pl line 17.\n#     Structures begin differing at:\n#          \$go
+t->{b} = Does not exist\n#     \$expected->{b} = '1'\n",
  std => "ok 1 - pass?\nnot ok 2 - fail?\n1..2\n",
  todo => undef,
}
[download]