First off, I'm not sure if "intangible" is standard terminology, Wikipedia defines an intangible object to be: "objects that are proved to exist, but which cannot be explicitly constructed". So if someone could point me towards better terminology, I'd appreciate it.
The linked article from Wikipedia claims that the axiom of choice implies the existence of such intangible objects, using non-measurable sets as an example. I was wondering if the law of the excluded middle gives us similar examples of intangible objects. If there are no such examples, is it possible to prove that all objects proven to exist using the LEM also exist constructively? Would that even be possible to prove?