Reading Wald's book (page 380, end of the first paragraph of section 14.1) while the author is giving an overall discussion of quantum field theories you can read

However, for the more interesting case of a theory with interactions (i.e., for a field or fields satisfying nonlinear equations) one is led unavoidably to consider the products of field operators at the same spacetime point. Such quantities have no natural mathematical meaning, and consequently, with the exception of some simple models in lower spacetime dimensions, there are at the present no known examples of quantum field theories of physically reasonable interacting fields which are mathematically well defined.

What does Wald exactly mean by mathematically well defined, which are the problems, for example in QED that make it ill defined?