how to tan a caribou skin

I finally tanned my caribou skin this weekend after putting off the project since early November. I did one last year, and remember the whole process being frustrating and hard – the kind of work that makes you physically sore. And it all needs to happen at once.
My goal this time was to meet the standard spelled out in John and Geri McPherson's Primitive Wilderness Skills, Applied and Advanced: the finished robe should "drape over your arm like a blanket, not fold like paper."
In my limited understanding of the process, actual tanning – with tannin, alum, or other chemicals – changes the chemical composition of the skin. Another, simpler way is to just replace the water in the skin with oil (also a chemical change, I suppose) and work the fibers as the skin dries.
That’s what I tried, partly with some caribou brains, which look like strawberry yogurt and are only safe because spongiform encephalopathy hasn’t been found in Alaska, and partly with a mixture of Ivory soap and neatsfoot oil.
I thawed out the skin, fleshed it, scraped off the membrane, shampooed and rinsed it, rubbed the brains and soap into it, and worked it hard on the stretcher.
Forty-eight hours later, it’s done. The skin is more like a blanket than paper, but still a little stiff, a little greasy, and not quite like store-bought. Not that you can buy a caribou skin. Caribou have warble flies that live under their skin at various times of the year, and the bugs leave unsightly marks. The store here instead sells tanned reindeer, the domestic version of caribou.
I can't help but wonder now if a turn in the dryer with some tennis balls wouldn’t soften up the skin. But there’s a good chance that would lead to disaster for the dryer or the fragile, hollow hairs, so I think I'll stick with what I've got.