By sheer good luck, as we were checking out hostals in Arica, Chile, a motorcycle cop pulled up, and in the course of conversation with him, mentioned the tire problem. He said that there was a shop in town that had a good selection. Riding a Japanese 400cc dirt bike, he led us through town at a frantic pace, complete with stopping traffic for us at intersections etc., to a small shop. They had about 20 tires in stock. European brand, their very biggest, was a little small, and a straight street tire, but would do, considering the alternatives.
Maps are never to be taken as the absolute gospel.
For southern Chile, the maps show a road almost to the very end of the country - there isn't one. The maps show the roads the government would like to have - not the reality. A road in Brazil that is on the map doesn't exist at all, another one isn't on the map, another one marked as paved is mud. A road in Malawi marked as the main route to a town is supposed to be paved - it's a dirt and rock road up a mountainside. And sometimes you get pleasantly surprised, as the road which you thought was gravel has been paved since the map or guidebook was published.