The more I think about it, the more it seems like sexual characteristics, and racial characteristics, fall into the same basket. Anthropology describes race by bone shape, just like sex is described by physical characteristics. DNA always tells the truth, but if someone with male DNA can claim to be female, then why can't someone from the Mongoloid race claim to be Negroid, especially since it's possible to alter the physical characteristics of bones with surgery?
That is, race is not a social construct. It is an anthropological classification, just like sex.
Racial Identification in the Skull and Teeth
Jodi Blumenfeld, University of Western Ontario Journal of Anthropology, 2011; pg 5
http://ir.lib.uwo.ca/cgi/viewcontent.cgi?article=1137&context=totem