This is the country's first honorary consulate in Croatia. Ireland has an embassy in Zagreb, which was opened in 2015.
Before the embassy was opened, Tedeschi was Ireland's honorary consul for the entire Croatia, and was appointed to that post in 2006.
He will now be in charge of the four Dalmatian counties.
Tedeschi said the role of the consulate was to assist Irish nationals in Dalmatia in matters such as the loss of personal documents. The consulate will also be open to Croatian nationals seeking information on Ireland.
Earlier in the day, Split Mayor Andro Krstulovic Opara received the incoming Irish Ambassador Olive Hempenstall, who presented her credentials to President Kolinda Grabar-Kitarovic in September 2016, as the second Irish ambassador to Croatia.
Krstulovic Opara said that the number of Irish tourists in Split grew 26% in 2016.
Hempenstall said that she was aware that a dozen Irish nationals lived in Split, having moved there because of their partners, and that they were very happy with their life there.
She said that so far 17,000 Croatian nationals had emigrated to Ireland, that they were all registered and had jobs.
Croatia and Ireland established diplomatic relations in 1995.