Great question, I asked it myself and here’s what I came up with. You can just use this in a custom plugin or in functions.php.
// turn on Output Buffering (hence *ob*)
ob_start();
// register a callback to run after WordPress has outputed everything
add_action('shutdown', function () {
// get the output buffer and store it to a variable
$html = ob_get_clean();
// setup the PHP native html beautifier called tidy
// note that you need to have libtidy installed
// but chances are that your server already have it
$tidy = new \tidy;
$tidy->parseString($html, [
'indent' => true,
'output-xhtml' => true,
'wrap' => 200
], 'utf8');
$tidy->cleanRepair();
// output the beautified html
echo $tidy;
// Use 0 as third parameter to the add_action so you will
// register the action as early as possible. Otherwise
// WordPress will flush the output buffer before you do...
}, 0);