That doesn't really matter. If all you care about is physical reality, you have to apply its implications on everything, and Gödel's incompleteness reappears in a different form. First, consider that undecidability is more general -- i.e., stronger -- than incompleteness, as undecidability implies incompleteness but not vice-versa. If you admit infinity, then you have undecidability and therefore incompleteness. If you don't have infinity because you decide to only care about physical reality, then you don't have undecidability, but then you must treat time as physical (and therefore finite) as well.

While you no longer have undecidability (and therefore no incompleteness, either), you do have its finite form, namely infeasibility (due to computational complexity), which, given an assumption that time is finite, amounts to precisely the same result. Almost every theorem about undecidability can be generalized to infeasibility. For example, instead of undecidability of the halting problem, we get infeasibility of the bounded halting problem (that's the basis for the hierarchy theorems, which gave birth to the notion of computational complexity, just as the halting problem gave birth to the notion of computability). Similarly, by bounded halting, instead of a consequence of incompleteness, i.e. "there are statements that can neither be proven nor falsified", you get "there are statements that can be neither feasibly proven nor falsified" -- this is a direct corollary of the hierarchy theorems -- and the problem remains the same. The only question is whether you're willing to admit infinity (or what kinds of infinity you're willing to admit) as a convenience, or as an approximation of large finite quantities. But choosing not to admit infinity doesn't really make anything simpler (on the contrary, things may get more tedious).