Suggestions for binding together popcorn into large, edible structures?

We ended up using straight corn syrup heated to soft ball stage. Two critical factors were (1) use the right proportions and (2) do the mixing/coating inside a big wok to keep it from cooling before adding to the mold.

Getting the right proportions takes trial and error. Too little corn syrup will not hold the popcorn together well and too much will make the mixture too sticky to press into the mold. We ended up using about 1/4 cup per batch of air-popped corn.

The biggest wok we could find was 18" but even bigger would have probably have been even better.

As @rumtscho wrote, you likely can't make puffed corn the way your popcorn cakes are made. You might want to look into recipes for popcorn balls or similar recipes for different kinds of candy-covered popcorn, which can help you achieve a "glue" that actually holds your popcorn together. The recipe here has baking soda, added at the end, that causes the entire pot of sugar to foam up and become less rock-hard once it cools. I've made recipes like that one before; they taste pretty good and the popcorn does hold together.

These cakes are made by expanding the grains in a mold under vacuum. You won't be able to replicate the process, unless you have the machine for it. I have not done research on that, but I suspect they are only produced for industrial purposes - so too expensive, too large, and too high capacity for a home kitchen.

A description from http://www.madehow.com/Volume-4/Rice-Cake.html of what the machine does, in the case of rice cakes (AFAIK, the maize process is the same):

The rice is gravity-fed from the hopper into the cast-iron mold or cooking head in the popping machine. The mold is heated to hundreds of degrees, and a slide plate opens to impose a vacuum on the moist rice mass. After 8 to 10 seconds of exposure to heat at this pressure, the lid of the mold expands, creating an even greater vacuum on the contents. In the last few seconds of heating, the mixture explodes to fill the given space.

You can of course continue gluing your popcorn with different substances and see what comes out. Probably the most effective way would be to take existing recipes for granola bars, use popcorn in the place of rice kernels, and see how well you like the result. The caveat is that it won't be very similar to the commercially made puffed grain cakes, you will always taste the glue. You will be making a different product, even if it is tasty on its own right.

Update If what you wanted was to have a kind of "edible styrofoam" to use as a sculpting material, this is commonly done with puffed rice and molten marshmallows, and should be similarly doable with popcorn, unless the oil interferes. There are lots of tutorials for how to do it, a top search result for me was https://chelsweets.com/cake-decorating-rice-krispies/.