Geodesic
Simultaneous decomplexification of a~pair of complex matrices via a~unitary similarity transformation
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIV, Tome 395 (2011), pp. 5-8

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Let $(A,B)$ be a given pair of complex $n\times n$ matrices and let at least one of these matrices be unitarily irreducible. An algorithm for verifying whether $A$ and $B$ can be made real via the same unitary similarity transformation is proposed and justified. 
@article{ZNSL_2011_395_a0,
     author = {A. K. Abdikalykov and Kh. D. Ikramov},
     title = {Simultaneous decomplexification of a~pair of complex matrices via a~unitary similarity transformation},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--8},
     publisher = {mathdoc},
     volume = {395},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_395_a0/}
}
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A. K. Abdikalykov; Kh. D. Ikramov. Simultaneous decomplexification of a~pair of complex matrices via a~unitary similarity transformation. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIV, Tome 395 (2011), pp. 5-8. http://geodesic.mathdoc.fr/item/ZNSL_2011_395_a0/