Setting cookies in WP REST API requests

I’m trying to use setcookie inside a WP REST request, but it isn’t working. The REST request runs properly and performs it’s other duties well, but it doesn’t set any cookies.

Here’s my example code, which is running on mywebsite.com

add_action( 'rest_api_init', function () {
    register_rest_route( 'my_auth/v1', '/auth_login', array(
        'methods' => array('POST'),
        'callback' => 'auth_login',
    ));

});


function auth_login( WP_REST_Request $request ) {
    update_post_meta(1234, 'test_field', 'test_value'); // this works!
    setcookie('auth_token', 'test1234', time()+3600, "/", 'mywebsite.com'); // this doesn't work
    return 'test';
}

If I send an AJAX request to my endpoint (mywebsite.com/wp-json/my_auth/v1/auth_login), the update_post_meta call works fine, but the setcookie call does not. I have tested this by visiting mywebsite.com after a request, which has no cookies set.