I had this problem. I used composer to install a local version of SilverStripe, and then I ftp it to my host. Locally everything worked, but on my host I had the "/index.php/" being appended to <% base_tag %>.
I solved it by editing my .htaccess file. In it there is the line:
RewriteBase '/some directory'
I removed this line from my host .htaccess file. And that solved it.