I always wondered why Italy did not stick with germany at the time of the Franco-Prussian war of 1870.
There are obviously a few reasons: the absolute priority was the recovery of Rome, and the new Italy was very worried about the possible reactions of other Catholic powers: given that France was busy with Prussia, and that Spain was not really in the position to intervene, the big question was Austria (actually, the interesting point of TTL would be if Austria were to come into the war, both for revenge against Italy and Prussia and to defend the Holy See: my guess id that it would not change the ultimate result of the war, but at the same time it could deflagrate into a continental war, much longer that OTL limited conflict).
Btw, Garibaldi participated in the war, leading a volunteer corps on the side of the French (which shows that garibaldi too was a very strange guy, considering that he was wounded - and his wife killed - in 1848 by french troops, and that french troops were stationed in Lazio to guarantee the Papal state). However, I would not expect any big ripple from the Commune: OTL, it did not make a big impact anywhere in Europe (compare with the widespread insurrections of 1848).
Just a note: Italy was surely interested in Nice and Savoy (which had been ceded to france just 10 years before the war), and probably Corsica too (I am not so sure that Corsica was such an issue in 1870).
Tunisia was not yet a French protectorate (it was occupied in 1881), and at the time italian interests in Tunisia were surely greater and more established than French ones. Possibly, one of the outcomes of the peace treaty might have been to recognize an Italian interest in North Africa east of Algeria.
Going back to the possibility that Austria might come in for another round, I would expect that this might mean an early end for the AH empire.
A possible outcome of the war might even have been Austria and Bohemia-Moravia joining the German states in the declaration of the empire; Hungary and Croatia declared an autonomous kingdom (under an Habsburg king??), and Italy grabbing Dalmatia, Istria, Trentino and South Tirol, with Slovenia made into a semi-autonomous principality under joint Italo-German protection.Russia might get Galicia and Austrian Poland, as compensation.
This would surely change the future.
OTOH, supposing AH stays put, I still would expect a strong tie between Germany and Italy, which might ultimately end up either in a cooperation on colonial issues (just a thought: France already had interests in SE Asia in 1870. The peace treaty did not mention any colonial settlement - Bismark was not very eager to go for colonial aggrandizement, he preferred to find an European lebensraum for Prussia
But it would surely be interesting a joint Italo-German protectorate over Indochina. During the late 1860's Italy was - quite shily - sniffing around those areas, as well as in the Red sea).
Ultimately, I would expect that the Italo-German alliance might decide to crack the AH nut (1890??). This might open very interesting scenarios: Russia might be a willing partner (a la partition of Poland
); France might be still recovering (and the peace treaty might have been worse). UK would not be amused for sure. Still, without a strong continental ally its option would be reduced...