Hello @Upndown
Welcome to the Wefugees community and thank you for your question.
There are a few question that need to be clarified first in order to answer your question:
1) You went to Italy on a tourist visa. So I assume that the visa already expired in the past and you are not holding a valid visa by any European country at the moment, right?
2) You didn't stay in Europe in the meantime, instead you returned to your home country or you left the EU/Schengen-area, correct?
If the answer to both of the questions above is "Yes" then there shouldn't be a problem with Dublin III in this case as far as I can see.
The Dublin III regulation comes into play when you've already been registered in another European country or if you received a visa/residence title/etc. by said country. As an example: If you go to Italy with a tourist visa and then go directly to the UK to claim asylum, they are going to send you back because Italy issued you a visa.
If you leave EU for more than 3 months (e.g. to go back to your home country) your case is "reset" from the point of view of Dublin III. For example: If you go to the UK now they can treat your application as a new case that is not related to your previous trip to Italy. See the answer of Marcel in this thread for further explanation: https://***.wefugees.de/102057/claim-asylum-for-second-time-in-eu.
Best regards,
Thorgen