Unfortunately asylees don't really have a physical "residence permit" or visa. We get an I-94 that says we're lawfully "admitted indefinitely" to the US. With that I-94 we obtain an unrestricted SSN, which allows us to work. And we get an RTD, which allows us to travel. But we don't have a catch-all document with all that information like LPRs or people with an actual visa stamped on their passport. The closest thing would be the RTD, which is only valid for one year (lol), but many people don't know what it is or how it's used. Your best best will be trying to explain this to the Italian Embassy people using USCIS documentation or websites. Many times the foreign embassies themselves will have something on their websites to this effect, but not all do. Bring everything you have - I-94, approval letter, RTD, work permit, etc. Good luck!