Is there an event related to the creation of a session cookie for anonymous users?

I need to set an additional cookie whenever an anonymous user received a session cookie. This happens when selecting a facet in Search API for example. The specific use case is that I need an extra cookie set specifically for a third-party integration (GTranslate). hook_user_login doesn't meet the use case because the user isn't actually logged in but they do receive a session cookie upon selecting a facet. Is there some event I can subscribe to when that happens so I can inject my custom cookie?

You can try out an event response event subscriber. For E.g: Create a file in CUSTOM_MODULE/src/EventSubscriber/SetSessionCookieSubscriber.php

namespace Drupal\CUSTOM_MODULE\EventSubscriber;

use Symfony\Component\HttpKernel\Event\FilterResponseEvent;
use Symfony\Component\HttpKernel\KernelEvents;
use Symfony\Component\EventDispatcher\EventSubscriberInterface;
use Symfony\Component\HttpFoundation\Request;

class SetSessionCookieSubscriber implements EventSubscriberInterface {
  public function onResponse(FilterResponseEvent $event) {
    /** @var Request $request */
    $request = $event->getRequest();
    // Check if this is the right request, for example posting facets
    $session = $request->getSession();
    // Check for session values
    $response = $event->getResponse();
    /** Your custom code goes here */
    $response->headers->setCookie(new Cookie(...));
  }
  
  public static function getSubscribedEvents() {
    $events[KernelEvents::RESPONSE][] = ['onResponse'];
    return $events;
  }
}

Register it in your services file CUSTOM_MODULE.services.yml

services:
  CUSTOM_MODULE.setsessioncookie:
  class: Drupal\CUSTOM_MODULE\EventSubscriber\SetSessionCookieSubscriber
  tags:
    - { name: event_subscriber }