You must sacrifice one or the other. As we mostly care more about consistency (we don't want to make nonsense statements), completeness is sacrificed. Meaning, there will be things we can write down that we won't ever be able to prove true or false (relative to a set of axioms).
There are many other implications and subtleties, like potentially not being able to show a set of axioms is complete or consistent, potentially not being able to show the equivalence of axioms, not knowing whether adding an axiom is still consistent, etc.
Turing understood Godel's incompleteness theorem and applied it to computation. The computer science way of saying the above is that one can never make a program that can determine whether other programs halt or not (in finite time). Practically, this means that there will never be a compiler that will tell you whether you program has a bug. This is called the Halting Problem [1] and directly relates to Godel's incompleteness theorem as one can cast mathematics through the lens of computation and running Turing machines.
Presumably the "classification" here is some combination of whether models are equivalent or consistent and complete.
[0] https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...