ROME — Pity the weak and wobbly new Italian government. While its incompatible coalition partners pretend to tolerate one another, they are relegated to being nervous bystanders, and potential casualties, in the real fight for power: the Battle of the Two Matteos.

Technically speaking, the former center-left prime minister Matteo Renzi and the popular anti-migrant nationalist leader Matteo Salvini may be out of power. But their sparring has come to dominate Italy’s political life and the debate over the country’s direction.

The two Matteos share more than just a name. Both are ambitious. Both are extremely talented men in their 40s who have spent their entire professional lives in politics. Each represents a strikingly different vision of Italy’s future. Most of all, they need each other to get back into power.

Mr. Salvini needs the once high-flying Mr. Renzi as a punching bag — mocking him as a “misunderstood genius” whose feats “Italians didn’t notice”— to maintain his Everyman appeal as he seeks to destabilize the government and strengthen his policies against immigrants and the European Union.