When straining things that clog up the screen, I tend to use a spoon or spatula to move the stuff around, so you can get expose un-obstructed parts of the screen.
Of course, you might end up pushing some of the pulp through the process, but it's generally minimal.
You can then either dump out the pulp as you go (possibly into muslin or cheese cloth, let it cool, then give that part a squeeze), or use the back of a large spoon or a rubber scraper to push the pulp against the strainer to get out some extra liquid. (or use an oven mitt, covered in a plastic bag ... it's messy, but it works).
Of course, if you have a cheap strainer, too much force on it can cause the whole screen to pop out, splashing you and making a quite large mess. (I've since made sure that my strainers weren't basically a screen just held in by pressure to a loop of metal that attached to the handle, but are actually one integral part).
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If you're looking at buying new equipment, I'd look into getting a chinoise with the proper sized stick (not sure what it's called), which you can use to basically compress all of the pulp into the bottom as you're working.
Best Answer
As noted in "History of Tofu and Tofu Products", alkaline earth metals (such as calcium and magnesium) in solution cause the proteins in soy milk to curdle. As I discovered by searching for "Magnesium in table salt" and "Calcium in table salt", a cup of salt contains about 3 mg of magnesium and 70 mg of calclum. While those concentrations aren't high enough to turn soy milk into full-on tofu, they should be enough to cause some curdling.
All of this hinges, however, on the soy milk being heated. There are a number of soy milk products available in the US (e.g. Silk) that contain added calcium for health purposes--primarily in order to compete with the calcium content of dairy milk--that, themselves, do not curdle because of the low temperatures of storage associated with consumer soy milk.
While some soy starch is contained in the curds of soy milk, the curdling reaction for any solution primarily involves the kinking of the proteins in that solution, trapping within them fats and starches, in order to form a stable matrix of such.