Enumeration of discrete objects for which recognition problem is coNP-complete
This is not a complete answer, but let me try to explain the case for the Hamiltonian graphs and the non-Hamiltonian graphs with $n$ vertices.

There are $2^{\binom{n}{2}}$ graphs on $n$ vertices, and each of them is either Hamiltonian or non-Hamiltonian.  So one of the following is true.


There are at least $2^{\binom{n}{2}}/2$ Hamiltonian graphs.
There are at least $2^{\binom{n}{2}}/2$ non-Hamiltonian graphs.


I don't know which is true, but suppose 1 is true.  Let $H(n)$ be the number of Hamiltonian graphs with $n$ vertices.  Then $H(n)\geq 2^{\binom{n}{2}}/2$.  In this case, listing all the Hamiltonian graphs with $n$ vertices can be easily done in output polynomial time: Look through all the $2^{\binom{n}{2}}$ graphs, check the Hamiltonicity for each of them (in $2^{O(n)}$ time), and output if it is Hamiltonian.  The running time is $O(2^{\binom{n}{2}}\times 2^{O(n)}) = O(H(n)^2)$, since $2^{O(n)} = O(H(n))$.  If 2 is true, then by the same argument we can list all the non-Hamiltonian graphs with $n$ vertices in output polynomial time.

Namely, if the output size is polynomial in $2^{\binom{n}{2}}$, then such a trivial algorithm runs in output polynomial time (assuming that the verification can be done in $\text{poly}(2^{\binom{n}{2}})$ time).  So the only interesting cases are when the output sizes are subpolynomial in $2^{\binom{n}{2}}$.