A little nail or a screw might work - maybe not to reinforce the whole prong but to definitely give you some reinforcement on the very tip of the point. Find the longest, thinnest round-head screw you can, screw one into each tip, then finish it with a little dab of clay or Apoxie Sculpt, gesso, and paint. You'll probably have to touch up the paint after each con, but the foam tips won't get quite so mangled if you rest them on the floor.
You could also go over the tips with a bit of paper mache which will give them a bit more surface structure. Use a heavy paper (brown paper grocery bag or card stock), use your geometry skills to lay out a shape that you can wrap into a cone that fits tightly around the trident tip, thoroughly soak it in your paper mache solution (probably just white glue and water), let dry, then gesso and paint. This would be sturdier than the foam alone, but you probably would still want to do extra reinforcement right on the tip (e.g. metal screw) because the paper mache may still crush when you set the tips on the ground.
Anything more than that (e.g. making a core out of MDF) will likely need you to rebuild the whole thing from scratch.