Consider a $2n$ real symplectic space - the usual $\mathbb R^{2n}$.
Suppose that the same space could be endowed too with an Euclidean structure, by which the vectors of the symplectic basis are orthogonal.
In Symplectic Geometry by A. T. Fomenko it is said that
Generally speaking, the connection between the Euclidean and the symplectic structures is destroyed by any linear non-singular transformation of $\mathbb R^{2n}$
I think that such "connection" is indeed preserved by transformations of the unitary group...am I wrong?