Was there a real Sweeney Todd?

Illustration from "The art of honing a razor. The art of shaving," 1884. Charles Lonergan/Public domain

Sweeney Todd is a well-known fictional character, a murderous barber who colludes with a cook to bake his victims into pies. There are many instances of the demon barber story being touted as a tale based in real-life events, but how true is that?