This question is related to the previous post, "A question about unitary and anti-unitary matrices". Following the suggestion of Lspice, I am posting it as a separate question, as it might be of independent interest.
Let $\cal{U}(n)$ and $\tilde{\cal{U}}(n)$ denote the subspaces of $\mathbf{C}^{n\times n}$ formed by the unitary and the anti-unitary matrices, respectively.
The question is: how separated are $\cal{U}(n)$ and $\tilde{\cal{U}}(n)$, when seen as subspaces of $O(2n)$?
More precisely, what is the value of
$d({\cal U}(n),\tilde{\cal U}(n))=\inf∥U-\tilde{U}∥$,
where the infimum is taken over $U\in \cal{U}(n)$ and $\tilde U\in \tilde{\cal{U}}(n)$, denoting by $∥U∥$ the operator norm of $U$ for the euclidean norm in $R^{2n}$?
It is shown below that $d({\cal U}(n),\tilde{\cal U}(n))\ge \sqrt{2}$. It would be interesting to know if an exact determination is known or can be obtained, and how it depends on $n$.