The policy violates German laws that give people the right to use pseudonyms online, said the data protection agency in Schleswig-Holstein.
The agency has issued a decree demanding that Facebook let people use fake names immediately.
Facebook said it would fight the decree "vigorously" and that its naming policy met European data protection rules.
"It is unacceptable that a US portal like Facebook violates German data protection law unopposed and with no prospect of an end," said Thilo Weichert, head of the regional data protection office in Schleswig Holstein, in a statement.
The ability to use a pseudonym on Facebook was "reasonable" said Mr Weichert and would allow people to use the service "without fear of unpleasant consequences".