I had done at least 50,000 miles of touring before I had an aluminum frame break in Australia in 2001 - so the overall incidence rate is fairly low. When that happened, I found the easiest way was to fly back to USA and pick up a spare bike.

While it wasn't third world, when I did a later expedition trip across Russia in 2007, I did do two things: (1) I left behind a spare Trek 520 in Russia on a pre-trip in 2006 and (2) I used a steel frame bike. So this way if a bike was broken or stolen, I could pick up my spare after a train ride within the same country.

So even though I did pick up a steel frame bike, I'd say risks of breaking a frame and not being able to repair it - would be pretty far down on the overall risk list for a third world tour and even lower on the list for a tour in the developed world.