All sides agree to establish a sovereign council by rotation between the military and civilians for a period of three years.
The president has already waded into the debate, saying Britain was "in somewhat turmoil" and that it was up to the people if she stayed in power.
UK PM May clings to power following the resignation of her foreign and Brexit ministers in protest at her strategy for leaving the European Union.
Lingering political tensions over the question of how to control immigration have broken out into the open, and the fallout is reshaping alliances and stoking old rivalries from Rome to Berlin, Paris and Vienna.
The leaders of the right-wing League and the 5-Star Movement patched up their alliance after agreeing to substitute a eurosceptic they had initially proposed as economy minister, a nomination that had been rejected by the head of state.
President Sergio Mattarella has called in Carlo Cottarelli after two anti-establishment parties angrily abandoned their plans to form a coalition in the face of a veto from the head of state over their choice of economy minister.