You could have a shot at this if you have
Garibaldi decide to set up a republic in Southern Italy during the
Expedition of the Thousand, as he was encouraged to do by Sicilian politician
Francesco Crispi. In OTL I think (not quite sure, to be honest) he had a secret agreement with Cavour to yield Southern Italy to the Savoys to unite the country, but if he reneges on the agreement (atypical of him, he was quite committed to the unitarian ideal) or if he doesn't deal with Cavour at all (possible, he was not fond of the man), he might go for a Southern Republic.
Having said that, I am not quite sure whether a republican regime could have lasted long. Other powers could have intervened "to quell the rebels", first of all the Savoys, and in their case they could take advantage of the situation to conquer the South anyway. And even if Garibaldi's regime survived this, I reckon that the country could have fragmented, with Sicily going for complete independence, even from Naples.
Finally, as much as I admire the man, I am not quite sure that Garibaldi had in him the stuff to build a stable, peaceful and popular Republican regime. He was quite the anticlerical, among other things, and the Church (together with the most ardent Catholics, of which there were many in the South) would be against him and his government. And what after he dies? Maybe his successors go for a proto-fascist authoritarian regime.