Given a bit part as a harem girl, Wanda lives out her fantasy. So does the self-regarding Sheik (a plump former barber). Still in costume, he takes his devoted fan to an outdoor cafe where, recognized by the patrons and throwing caution to the wind, he sweeps her into a slow tango, whistling seductively in her ear.

Back in Rome, Ivan is engaged in his own subterfuge — covering up Wanda’s absence to his proper relations. “The White Sheik” is full of parallelisms. At various times both Ivan and Wanda are considered to be crazy. Ivan is as concerned with outward appearances as Wanda is inwardly directed. His desire for an audience with the pope rhymes with hers to meet her idol. The performers lunch on the beach, while Ivan dines in a restaurant with his bourgeois family. Fellini cuts from the Sheik’s failed seduction to enthusiastic applause in the opera house where Ivan attends a performance of “Don Giovanni.”

“The White Sheik” did poorly in Italy, though it was better received a few years later in New York. By then, Fellini’s reputation preceded him. Writing in The New York Times, Bosley Crowther found the movie “surprisingly broad and ingenuous, in the manner of early silent comedies,” identifying Sordi as “a student of the Mack Sennett school.”

Integrating aspects of fantasy into a documentary backdrop, “The White Sheik” is an example of what would be called “rosy” neorealism. Several of these films, among them Visconti’s “Bellissima” and Antonioni’s “The Lady Without Camelias” (both made in the early ’50s) similarly concerned the entertainment industry and, as in “The White Sheik,” had actors playing “actors.” Fellini pushes this strategy the furthest. The hapless Sheik’s row with his harridan wife becomes an entertaining show for the production’s cast and crew. Rome itself is inhabited by performers. A vivacious prostitute (the adorable Giulietta Masina, Fellini’s wife) shows up and cuts a few capers in an empty piazza — in effect providing a trailer for the director’s later “Nights of Cabiria,” in which she would appear as the same character.

“The White Sheik” anticipated or inspired a number of later Hollywood comedies, including Gene Wilder’s “The World’s Greatest Lover,” Neil LaBute’s “Nurse Betty” and especially Woody Allen’s “The Purple Rose of Cairo.” But farcical as it is, “The White Sheik” has a serious interest in fandom’s quasi-religious pathology. The movie’s spiritual twin is Billy Wilder’s “Sunset Boulevard,” made two years earlier and similarly self-reflexive in pondering the effect of the so-called dream factory.