It's the local branch of the public health service in your area. The ASL handles SSN enrolment, GP assignments, vaccinations, and basic public-health services. It's where you go to sign up for the health system after you arrive.
You can't get a tessera sanitaria, pick a doctor, or use most public health services without registering at your local ASL. It's also where you book vaccinations and get health certificates.
"On her second week in Ravenna, Aiko went to her ASL with her permesso, codice fiscale, and lease contract. She picked a pediatrician for her son from a printed list and walked out enrolled in the SSN."
Each region calls them slightly differently. In Lombardy they're ATS or ASST, in Veneto they're ULSS, but the function is the same.
Official source
Ministero della Salute
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