The notion of Fixed establishment (FE) is a highly controversial topic in EU VAT.
The concept is extremely import since it affects crucial aspects of the EU VAT legislation, such as the place of supply (in which jurisdiction VAT needs to be paid), the debtor of VAT (who is liable to pay VAT / reverse charge), VAT recovery (what refund procedure to follow for obtaining VAT credits), etc.
The definition of a permanent establishment for VAT purposes must not be confused with the PE concept for direct taxes. Although certain jurisdictions by default expect a VAT FE once a corporate tax PE is recognized, to test the existence of a PE for direct taxes, other criteria apply!
Source VAT Consult