Links are rendered without the subdirectory path for anonymous users on multisites hosted in subdirectories

Here is my setup:
A Drupal 8 multisite installation on a subdomain that can be reached with:
http://my.subdomain.be/mysite

If I am logged in as an administrator or with any other available role, everything is fine.
But
If I browse my site as an anonymous user, then the links of the menus doesn't include anymore the "mysite" path.
This is true for all the menus even the "user account menu"

Exemple:
The link to logout is http://my.subdomain.be/mysite/user/logout when I am logged in (which is fine) but the login link is http://my.subdomain.be/user/login (which is really bad as "/mysite" is missing)

Of course, all my hand-made links have also this problem.

I don't think it is a setup problem on my side because it is working when logged in (but I could be wrong)
In any case, here are the only changes I have made to htaccess: (uncommenting the following 3 lines)

  # To redirect all users to access the site WITHOUT the 'www.' prefix,
  # (http://www.example.com/foo will be redirected to http://example.com/foo)
  # uncomment the following:
RewriteCond %{HTTP_HOST} ^www\.(.+)$ [NC]
RewriteRule ^ http%{ENV:protossl}://%1%{REQUEST_URI} [L,R=301]

  # Modify the RewriteBase if you are using Drupal in a subdirectory or in a
  # VirtualDocumentRoot and the rewrite rules are not working properly.
  # For example if your site is at http://example.com/drupal uncomment and
  # modify the following line:
RewriteBase /mysite

Here is my very basic apache2 setup:

<VirtualHost *:80>
        DocumentRoot /var/www/html/drupal8
        ServerName my.subdomain.be

        ServerAdmin web-master@subdomain.be
        ErrorLog /var/log/apache2/error/error_mysubdomain.log
        # Possible values include: debug, info, notice, warn, error, crit, alert, emerg.
        LogLevel error
        SetEnvIf Request_URI "\.(jpg|xml|png|gif|ico|js|css|swf|js?.|css?.)$" DontLog
        CustomLog /var/log/apache2/access/access_mysubdomain.log combined  Env=!DontLog
</VirtualHost>

Some other Apache config for Drupal:

<Directory /var/www/html/drupal8>
        Options FollowSymLinks
        AllowOverride All
        Require all granted
</Directory>

Some settings in settings.php:
(I have tried with and without the $base_url)

$base_url='http://my.subdomain.be/mysite';
$settings['trusted_host_patterns'] = array('^my\.subdomain\.be$');

And finally, for now, my sites.php file exists but contains nothing (I have tried with and without settings in it)