As shown by this post, a graphs is not determined by its multiset of spanning trees.
In fact, the two graphs below have the same multiset of spanning trees, but are non-isomorphic.
Lets call such graphs tree-equivalents.
Notice, however, that their complements are not tree-equivalents. In fact the following graph appears $5$ times in the multiset of the left graph's complement, but $11$ in the multiset of the right graph's complement.
So my question is: are there two non-isomorphic tree-equivalent graphs whose complements are tree-equivalents as well?
I understand that might be a hard question, so I would also like to know what properties graphs satisfying this condition must share.
For instance, it's easy to show that the maximum degree of the graph is the maximum of the maximum degrees of it's spanning trees, so two tree-equivalent graphs must share their maximum degrees. If their complements are tree-equivalent as well, then that must also be true for their minimum degree.
An analogous argument can be made to prove that two tree-equivalent graphs must share the length of its maximum path, but I don't know if this statement implies something nice for its complements, like in the previous case.