The primary evidence against a more complete theory are the experimental tests of Bell's Inequality. The experiments are compelling but have yet to simultaneously close the two major loopholes of timing constraints and detector efficiency. No experiment can conclusively eliminate the possibility of local but superluminal explanations. These are of interest for discrete models because these generally require that the velocity of propagation of a wave be significantly slower than the fastest possible causal effect.
Locality and discreteness are independent questions. However, locality, is the most powerful simplifying assumption in physics. If it is violated than the search for simplicity that motivates interest in discrete models is called into question.
Quantum mechanics was developed by theoreticians and experimenters working together. Results from each side influenced the other. It is inconceivable that a theory as strange and remarkable as quantum mechanics could have emerged as a complete theoretical structure that only needed verification as was the case with special and general relativity.
Developing a more complete theory will almost certainly require a similar collaborative effort of experimentalists and theoreticians feeding each others understanding and intuition. If the more complete theory requires `digital' space-time than this is almost certainly true. It may also require engineers to build the computers made possible by new physical insights that in turn allow theoreticians and experimenters to explore new possibilities.
Thus I call the outline of my ideas about how a more complete theory might look a digital physics fantasy. This is a section of the online book What is and what will be. This book assumes a discrete, finite but potentially infinite universe and makes the simplest possible assumption about the relationship between physical structure and conscious experience. It develops the implications of those assumptions in a way that connects to many spiritual and religious traditions without assuming any reality but that of the physical universe.