After getting the runaround between various departments of the Civil Registry and Ministry of the Interior (neither of which had any idea where I actually needed to go to get my DNI), I eventually found the tent in February.
I also went straight to one of the desks with no wait. I told them I wanted a 'turno' to apply for a DNI. They asked me what documents I had on me and I triumphantly produced everything you can possibly think of, all officialy translated and notarised as well as notarised copies of everything and a dozen perfectly framed photos with me gazing off to the left. They said they only needed to see my permanent residency visa and passport and they didn't need copies of anything. They took my fingerprints electronically, took my photo themselves, asked for 50 pesos and said my DNI will arrive in the post within 5-60 days.
No turnos, no fuss, all over in less than 15 minutes.
DNI hasn't arrived yet, but it's only been 46 days!
Al xx