I've also been known to do phantom aerobars on occasion. Pity there's no way to do a phantom "Obree tuck".
My position on one of my bikes has developed into an (admittedly naff!) version of the Obree position.
For a touring bike, I ride an early 90's rigid steel mtb. It's miles too small for me, with loads of seatpost showing such that the handlebars are miles lower than the saddle. I can get into a position with a pretty much flat back, and bar ends (fitted to normal flat bars) make it such that it is comfortable to ride with my arms folded up and tucked under my body.
Whilst it's not as quick as a road bike, it's a surprisingly fast bike for what it is - I think a lot of the speed is due to the position. Bear in mind it weighs a ton (over 16kg), has all the usual sort of tat associated with touring on it (mudguards, front and rear rack, lights etc) and has little 26" wheels with 1.5" wide (but slick) tyres on it. It's only a 7 speed too.
At one point my road bike was broken, so I used it to commute to work for a week. I was in very good form at the time, but I was surprised to find that it wasn't much slower than on my road bike over the same route. I think most of rides were around 19mph average, with the best commute of the week being an average 20.8mph over 15 miles. Yes - there was a bit of a tailwind, but it's not net downhill and there are plenty of junctions, lights etc to contend with, and the bike was still pretty quick on other days without the tailwind.
I doubt the bloke I overtook on a TT bike wearing full aero kit was that impressed, especially as I had a weeks worth of work shirts on the rear rack.