However state television RAI said in a documentary that the abbey was an inaccessible ruin closed to the public.
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In announcing its intention to revoke the permission, the ministry also cited violations of various contractual obligations including a failure to pay concession fees and do maintenance work.
Harnwell's institute appealed to a local administrative court, which blocked the eviction order, effectively allowing the organization to keep managing the monastery.
A ministry spokesman said it has now asked the State Council, Italy's top administrative court, to review the local court decision.
In a separate action, Italy's Court of Auditors said the institute did not pay rent of around €200,000 ($236,340) for 2018 and 2019.
Harnwell did not directly address this accusation in responding to questions from Reuters but said Italian authorities were trying to undermine the institute because of politics.
In text message exchanges with Reuters, Harnwell said the institute "never participated fraudulently in the tender as alleged", and argued that "the Ministry for Culture annulled its lease out of political considerations."
He confirmed Italian media reports that prosecutors in Rome are also conducting an ongoing probe of the institute.
Harnwell argued that the "criminal court is proceeding with its own case to discuss precisely the same material that we've just been cleared of by the administrative court."