Most likely diagnosis? Ascaris lumbricoides
Without being able to examine the worms more closely, you can still make a preliminary diagnosis based on the size and shape of the worms - A. lumbricoides is the largest roundworm (Nematode) that parasitizes humans. I'm guessing that this fact (and their resemblance to a bowl of noodles) prompted Patty to write "Nothing gives me the gag reflex quite like ascaris worms".
The sheer number is also consistent with the diagnosis - Ascaris adults can number in the 100's or even 1000's in infected individuals! Since they live in the small intestine, it's easy to see how intestinal obstruction can occur with heavy infections.
How are these worm acquired? Through ingestion of eggs in contaminated food and water.