The technique literally uses smoke in scenes to diffuse lighting.
I found this description on the director Alan Parker’s own website:
“The European style of single source, diffused light — dubbed ‘Le Lumiere Anglais’ in France, was also a puzzlement to the American crew. For many years in Europe, we had used smoke in scenes to diffuse the light. Traditionally incense burners were used to puff the soft smoke into the air to catch the shafts of light, but this presented problems for some of the New York crew who were steadfastly set in their ways.”
He further explains the complications that came along with the use of this technique specifically in the filming of Fame.
“Firstly, the props department and the lighting crew couldn’t agree on who was actually responsible for this new task of igniting the pellets and puffing away with the small hand-bellows before each shot. Secondly, a number of the crew (and actors) were appalled by the presence of the smoke and complained to their unions about the “noxious” working conditions. We were duly visited by representatives from the SAG and Local 644 who promptly called a halt to filming in conditions ‘possibly hazardous to their members’ health.’ We pointed out that we had been using incense on sets in England for many years, but not nearly as long as the Catholic Church who, with the exact same smoke, had according to the New York crew, apparently been asphyxiating their congregations for two millennia.”
So that answers that, I guess? But I have never heard of “Le Lumiere Anglais” before.
I am self-answering this question but am open to more details if anyone out there knows more and can share them in another answer.