The question and answer in:
If marginal probabilities equal, can we say anything about joint distribution?
gives an example such that one can have two different joint distribution from the same marginal distributions. The example presented considers random variables with different covariances.
But can one have a different joint distribution from the same marginal distributions with the same covariance?
I believe it cannot be Gaussian, as up to the second moment would characterize the Gaussian measure.
But in general, only the marginal measure and the covariance sounds too weak to characterize the joint distribution (but I might be mistaken).
Related, this paper mentions a difficulty of determining the joint distribution from marginals.
https://www.jstor.org/stable/pdf/2346074.pdf?refreqid=excelsior%3A41796b29d4c4cded5326d24523e3fede
In sum, my question is that do we have an example of pairs of random variables such that they have same marginals and the same covariance, but different joint distributions?