The possible Italian connection for Mark Anthony in about 1670 is uncertain.Some references in Ancestry.com show Mark's father as Edward Anthony, b 1654 Italy, d unknown, with Mark born at Genoa.I have not seen any documentation on this.This Edward Anthony would trace back to Derrick Anthony, whose line connects up to Anthony branches who came across into Rhode Island.
Tracing these Jewish Anthony lines in England is difficult because they were forced to hide their Judaism from the authorities.Jews were legally excluded from England dating back to the Edict of Expulsion in 1290.They were not legally permitted to live there until the law was changed under Cromwell in 1656, and did not achieve full equal rights until 1890.
Jews obviously did live in England before 1656.See the Jewish musicians imported from Venice by Henry VIII, and the goldsmith Anthonys who served the Crown during the brief years of Edward, and then under Queen Mary and Queen Elizabeth I.Whether these people maintained their Judaism in secret is not documented, obviously, but they probably did.It is known that some of them used clearly Jewish names in secret.For example, the musician you mentioned (trombone player John Anthony Bassano) died in prison.In his Will, he used the name Moyses [Moses].Incidentally, he died with no children and so he cannot have been a direct ancestor of our Mark Anthony.
When any non-Englishman entered the country (or re-entered after a long stay abroad), they were required to apply for visas of a sort.Jewish people can sometimes be spotted in these records because officials used the euphemism 'dirty and Eqyptian' to skirt the law of Jewish exclusion, probably from bribes or official pressure from above.Derrick Anthony was described as 'dark, dirty and Egyptian' on his visa document, a clear reference to his Jewish ancestry.
The possible reasons for presumed father Edward Anthony's presence in Italy could have been several.He was related to the goldsmith Anthonys which in turn would imply a logical connection to the banking industry.Mark himself alluded to banking in Genoa in his falsified pirate story.Some of the Anthonys are known to have gone abroad during the period.The son of mint director Derrick Anthony, named Francis, was a medical doctor who was expelled from England for selling an elixir made of a gold compound which poisoned many people.He went to the court of the Holy Roman Emperor in Germany/Austria. Possibly some of that group drifted down into Italy.Because of the secrecy required, however, records are scant or missing, and so these remain speculations at this point.