Answer: Fasciolopsis buski, the intestinal fluke
Although this is an older case (Case of the Week 227), I thought it was worth showing it again for the newer readers who may not have seen it. This is a parasite that we don't often see in the United States, but is common in Asia where freshwater plants are ingested.
The first step in making the diagnosis is recognizing that these objects are flukes - flat 'leaf-like' worms with an oral and ventral sucker:
Thank you to everyone who wrote in on this case!