What do Feynman diagrams mean?

Speaking of Freeman Dyson, reminded me an old debate between him and Feynman. What do the Feynman diagrams really mean?

Do Feynman diagrams in particle physics represent real processes happening there in superposition as Feynman claimed or are they merely a bookkeeping tool for the perturbation calculations? There are Feynman diagrams in condensed matter physics and even in fluid mechanics (called Wyld diagrams) which certainly does not reflect any superposition according to my understanding (at least not in quantum sense). Also loop diagrams always confused me. Can such processes be "real"? I don’t think that any collider experiment probed those virtual processes, only outgoing particles are observable.

So, what do you think? Are they real or just calculation tools? Does it matter? Is it possible to discriminate or should we “shut up and calculate”? Please leave your comments.