Supervisory authority and extra-territorial provisions

Answer:

Where the extra-territorial provisions of the EU GDPR apply, the controller or the processor must appoint a representative. That representative must be based in a Member State in which the relevant individuals are based. There is a limited exemption to the obligation to appoint a representative where the processing is occasional, is unlikely to be a risk to individuals and does not involve large scale processing of sensitive personal data.
So, there is this obligation to have a representative in the EU and the representative will have to face the relevant supervisory authorities and accept liability for breach of the Regulation, which could now be substantial.