Okay. The instructions in the link I posted above said you could, but I guess they're just wrong.
Now we're running into another, weirder problem. I went ahead and built a WYSIWYG page with the form and the form population cloud component on it. This landing page then gets pulled through an iframe into another site. When I pull it up on my computer the form is populated as it should be. But when I have others pull it up on their computers (people who most definitely have Eloqua cookies on their system) either get blank form fields or the form fields are getting populated by completely wrong data.
My frustration level is very high................