Non-EU citizen in Spain with a valid Italian Tourism Schengen Visa and applying for PhD in Spain, What do I do?

It doesn't look good.

I would try asking at the oficina de extranjería (or, if none exists where you are, the local police station) whether they can extend your stay with a temporary residence permit, but normally Schengen short-stay visas are supposed to be extended only in case of emergency, so probably they cannot.

They may have another solution, but I wouldn't give that a high probability, either.

You can also try going to another country, closer than your home country, and applying for a Spanish visa there; the country you should go to would depend on your nationality and whether you already hold any other visas. This is also unlikely to succeed, unfortunately, because of the rules that require people to apply in their place of residence. It is possible for exceptions to be made when a reason for travel arises after you have left home, and this certainly seems to qualify, but in practice it appears to be difficult to get consulates to acknowledge this possibility. It is probably even more difficult to get the third-party processors that are increasingly used to acknowledge it.

Your best bet is probably to talk to the university, explain your visa situation, and see whether they can either allow you to register while you are outside of Spain or at a different time.

The situation you describe should make it possible to request a Spanish visa while in Italy. Usually people should apply from their place of residence, but the rules allow exceptions when plans change on short notice.

I don't know if the Spanish embassy will agree that yours is a hardship case, and if they will process the application fast enough.