ROME — When he was prime minister of Italy, Matteo Renzi, fighting desperately to win a referendum on which he had staked his job, thought that one particular rally had gone well.

He was surprised, then, to discover that RT, the international Russian television network with close ties to the Kremlin that was formerly known as Russia Today, had mischaracterized the pro-Renzi event on Oct. 29, 2016, as “Protests against the Italian prime minister ahead of the referendum on the Constitution.”

Mr. Renzi’s opponents quickly latched on to the report and shared it widely on social media, prompting the Italian leader to express his displeasure directly to Vladimir V. Putin, the Russian president.

“Does it seem reasonable to you that Russia Today often uses headlines that are not true?” he asked Mr. Putin in a phone call. “Why do they have to have reports today on some protest against me, if that square is full of my people, defending our reform?”