Geodesic
Factoring birational maps of rational surfaces over the field of real numbers
Fundamentalʹnaâ i prikladnaâ matematika, Tome 3 (1997) no. 2, pp. 519-547

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In this work the classification of non-singular projective rational surfaces over the field of real numbers is given, the theorem about factoring of every birational map between this surfaces into elementary ones is proved, a full system of relations between the latter is found. 
@article{FPM_1997_3_2_a10,
     author = {Yu. M. Polyakova},
     title = {Factoring birational maps of rational surfaces over the field of real numbers},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {519--547},
     publisher = {mathdoc},
     volume = {3},
     number = {2},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1997_3_2_a10/}
}
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Yu. M. Polyakova. Factoring birational maps of rational surfaces over the field of real numbers. Fundamentalʹnaâ i prikladnaâ matematika, Tome 3 (1997) no. 2, pp. 519-547. http://geodesic.mathdoc.fr/item/FPM_1997_3_2_a10/