Been reading the book, "1491", and he discusses how many societies in North and South America developed "milpas", or self repairing farming techniques, that allowed for the repeated use of the same land for crops without the necessity of crop rotation and the not planting of particular plots of land. I know that my knowledge is limited so perhaps I'm not aware of it, but is there any record of that type of integrated farming on record in the Europe/Asia/Africa continents?
Two words: animal power.
Before I say more, I will add the caveat that my answer refers almost exclusively to European agriculture. I can't speak to whether or not milpa-type systems existed in Asia or Africa.
European agriculture was dominated by the use of draft animals and plows from a very early time. The so-called "Celtic" field system dates from the Bronze Age, for example. Although the types of draft animals, plow designs, and crops planted varied, along with other technologies such as crop rotation, integrating manure, etc., the basic mechanics of field agriculture in Europe had to do with the efficiency of animal power, which in turn meant trying to increase the speed of plowing (moving from oxen to increasingly better breeds of draft horses) and, by extension, the organization of farm systems into linear spaces which could be quickly plowed, planted, harvested, etc. The more land that could be planted per farmer, or per draft animal, meant increasing the chances of seeing a surplus for that year.
When you're planting by hand, however -- as the agricultural cultures of the First Nations did -- you have to find your efficiencies elsewhere. Here, intensive versus extensive use of land will arguably produce higher yields, and since the method of planting doesn't have to be uniform, it can take advantage of specific relationships between plants on a small scale.
This is not to say that regenerative practices were totally unknown in Europe, they were simply implemented on a different scale. Rotating between fallow pieces of land was commonplace in ancient times, as was the use of manure. I'm not sure that we have any ancient sources for what we'd call "companion planting" today, though -- I would be interested if anyone could find any.
I really love "1491", and I think that Mann actually addresses this in the chapter on milpas. I can't speak to Old World examples of integrated farming, but I think that Mann offers a pretty good implicit explanation as to why milpa style farming was common in America while was significantly less widespread in the Old World.
He makes reference to and goes in depth on how the food crops developed in the Americas were different from those of Eurasia in that they were not grasses like wheat or barley. While ancient humans in Mesopotamia or Turkey or wherever looked out on "landscapes of food" - entire fields packed with stalks of grain - the ancient humans in Central America had nothing comparable. On pages 199-200, he talks about how in ancient times wild grain fields grew almost as thick as modern farm fields throughout Eurasia. Eurasian peoples had "landscapes of food" as part of their "mental furniture" while Mesoamerican peoples did not (200). They created an "environment" to put their new crop in.
Furthermore, these wide fields worked. Considering that there were still high populations in Eurasia before the introduction of American crops, this method of farming worked, if not as well as the milpa. Why change something when you don't need to?
Source: 1491 by Charles C. Mann