It is theoretically possible in the 19th century.
Tunisia hosted a fairly significant immigrant community of Italians (mostly but not exclusively from Sicily) back then.
An earlier Italian unification could easily do the trick.
A different Italian policy after OTL-ish unification may also do, if France is kept out long enough.
Indeed, an easy POD would be something to the effect that the Bitish don't get Malta in the Napoleonic Wars, or don't keep it after Vienna (maybe a more succesful Kingdom of Naples in that timeframe).
Assume little butterlies into Risorgimento, and Tunisia would become a relatively easy target for Italian puppetization/expansionism in the 1870s.
The problem is that, of course, so soon after a struggle of national liberation, there would be very strong objections to the subjugation of other "nations" (as it was the case with East Africa IOTL) even if said nations will be widely regarded as "barbaric" (regardless that the Tunisian Constitution at the time, IOTL, was more progressive than the Italian one).
The progressive public opinion in the 1880s in Italy nourished a very principled opposition to colonialism. In the end, it did not work, but it was active, and argued very convincingly that colonies were against the notion of everything Italy stood for in the first place.
It took the coming to power a new, more nationalistic generation in the 1890s to get the whole colonial thing fly, and the "loss" of Tunisia (i.e., French takeover) was a major factor in that change of mind.