if ($this->getStatusCode() == 401)
        $this->setHeader('WWW-Authenticate', 'basic');

<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:ObjectManager/etc/config.xsd">
    <preference for="MagentoFrameworkWebapiRestResponse" type="SduRewriterRestResponse" />
</config>

<?php
namespace SduRewriterRest;


class Response extends MagentoFrameworkWebapiRestResponse
{
   public function _renderMessages()
   {
       $responseHttpCode = null;
       /** @var Exception $exception */
       foreach ($this->getException() as $exception) {
           $maskedException = $this->_errorProcessor->maskException($exception);
           $messageData = [
               'message' => $maskedException->getMessage(),
           ];
           if ($maskedException->getErrors()) {
               $messageData['errors'] = [];
               foreach ($maskedException->getErrors() as $errorMessage) {
                   $errorData['message'] = $errorMessage->getRawMessage();
                   $errorData['parameters'] = $errorMessage->getParameters();
                   $messageData['errors'][] = $errorData;
               }
           }
           if ($maskedException->getCode()) {
               $messageData['code'] = $maskedException->getCode();
           }
           if ($maskedException->getDetails()) {
               $messageData['parameters'] = $maskedException->getDetails();
           }
           if ($this->_appState->getMode() == MagentoFrameworkAppState::MODE_DEVELOPER) {
               $messageData['trace'] = $exception instanceof MagentoFrameworkWebapiException
                   ? $exception->getStackTrace()
                   : $exception->getTraceAsString();
           }
           $responseHttpCode = $maskedException->getHttpCode();
       }
       // set HTTP code of the last error, Content-Type, and all rendered error messages to body
       $this->setHttpResponseCode($responseHttpCode);
       $this->setMimeType($this->_renderer->getMimeType());
       if ($this->getStatusCode() == 401)
           $this->setHeader('WWW-Authenticate', 'basic');
       $this->setBody($this->_renderer->render($messageData));
       return $this;
   }
}