= Solution
Let $J_0$ be the standard <complex structure> on $\mathbb R^{2n}$, let $\omega_0$ be the standard <symplectic form>, and let $g_0$ be the Euclidean <inner product>. They satisfy
$$
g_0(u,v)=\omega_0(u,J_0v),
\qquad
\omega_0(u,v)=g_0(J_0u,v).
$$
The <two-out-of-three property for unitary structures> says that a real linear map preserving any two of these structures preserves the third. In terms of the corresponding <matrix groups>,
$$
GL(n,\mathbb C)\cap Sp(2n,\mathbb R)
=Sp(2n,\mathbb R)\cap O(2n)
=O(2n)\cap GL(n,\mathbb C)
=U(n).
$$
For example, if $A$ preserves $J_0$ and $\omega_0$, then
$$
g_0(Au,Av)=\omega_0(Au,J_0Av)
=\omega_0(Au,AJ_0v)=g_0(u,v),
$$
so $A$ preserves $g_0$. If it preserves $J_0$ and $g_0$, the second displayed identity shows that it preserves $\omega_0$. Finally, $J_0$ is uniquely determined by $g_0(J_0u,v)=\omega_0(u,v)$, so preservation of $g_0$ and $\omega_0$ implies $AJ_0=J_0A$. The common intersection is therefore the <unitary group>.
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