It is possible to request a referral within the registration page for new users. However, it isn't possible to require or force the referral in order to register.

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Attached is a Hooks file, once you import the attached file to the Hooks plugin system you will be able to:
  • Require new users to fill the referral field within the registration page.

If you want to add multi language support for the error message, just add the force_referral_empty language key to your language file, for instance adding the following to the global.lang.php file should work:
$l['force_referral_empty'] = "A referrer is required to register, please fill the referral field.";

If you don't want to use multi language support, you can simply edit the hook after importing it in order to edit the error message.