Hello, @martinzeltin !
Are those Laravel "projects" sharing the same domain? In this case, I believe you need to use Redis driver instead of the default file session. Look for Redis sessions in the docs.
As I don't have any idea about how your web server is configured, it is hard to say if it is related to a wrong Apache configuration, for example. In some cases, you need to correctly use mod_rewrite to make it works well with Apache. I do not recommend you to edit your .htaccess to use RewriteBase. So, check it too.
Make sure the APP_NAME is properly set for each "project" because the SESSION_COOKIE will rely on this.
Also, check the following in the config/session.php:
/*
|--------------------------------------------------------------------------
| Session Cookie Path
|--------------------------------------------------------------------------
|
| The session cookie path determines the path for which the cookie will
| be regarded as available. Typically, this will be the root path of
| your application but you are free to change this when necessary.
|
*/
Hope this helps. Let me know.