Here's another example. By a "computable well ordering" I will mean an index for a well-founded computable (total) linear order on $\omega$. Because ZFC is an effective theory, there must be some computable well ordering $\zeta$ that ZFC does not prove is a well ordering. This is because:
The set of indices of computable well orderings is strictly $\Pi^1_1$ but, because ZFC is an effective theory, the set of computable linear orderings that ZFC proves are well founded is $\Sigma^0_1$. So it cannot be true that a computable linear ordering is well founded if and only if ZFC proves it is well founded.
ZFC is a true theory, so if ZFC proves that a computable linear order $L$ is well founded then $L$ really is well founded.
Because ZFC does not prove that $\zeta$ is a well ordering, there is some model of ZFC in which $\zeta$ is either not a total linear ordering, or $\zeta$ is not well founded. Either of these can only happen in a non-well-founded model of ZFC, since by definition $\zeta$ really is a well ordering.
Although this type of example is related to the type from the question, the proof method sketched here does not make use of consistency statements, so I feel this counts as an essentially different example of the same underlying phenomenon.