Can two unitary similar real matrices be orthogonal similar?
suppose $A=U^tBU$ where $U$ is unitary, does there always exists a real orthogonal matrix $O$, such that $A=O^tBO$ ?



Can two unitary similar real matrix be orthogonal similar
Unitarily similar real matrices are always orthogonally similar. 
The proof can be found in Rached Mneimné and Frédéric Testard's book "Introduction à la théorie des groupes de Lie classiques". 


First of all, one sees that two complex square matrices $A$ and $B$ are unitarily similar
if and only if the pairs $(A,A^\star)$ and $(B,B^\star)$ are similar over the complex numbers. 
The direct implication is straightforward; for the converse, assume that we have an invertible matrix $P$ such that $A=PBP^{-1}$ and $A^\star=P B^\star P^{-1}$. 
Applying $(-)^\star$ to the second equality, we get $A=(P^\star)^{-1}BP^\star$, 
and using the first equality we deduce that $P^\star P$ commutes with $B$. 
Writing the polar decomposition of $P$ as $P=UH$, this means that $H^2$ commutes with $B$, 
which, as $H$ is a polynomial of $H^2$, yields that $H$ commutes with $B$. 
Therefore $UBU^{-1}=PBP^{-1}=A$, QED. 


Now, with the same line of reasoning, one shows that two real square matrices $A$ and $B$ are orthogonally similar if and only if the pairs $(A,A^t)$ and $(B,B^t)$ are similar
over the real numbers. 


One concludes by noting that if two pairs $(A\_1,A\_2)$ and $(B\_1,B\_2)$
of real square matrices are similar over the complex numbers, then they are also similar over the real numbers: this is a widely known special case of the Noether-Deuring Theorem
(for a classical elementary proof, see section 2.2. of [Invariance of simultaneous similarity and equivalence of matrices under extension of the ground field](http://arxiv.org/pdf/0902.4528v4.pdf)). 


