FOM: computable non-orderable abelian group?

[Kanovei]

There is a curios complementary question: 

is there a computable torsion-free abelian groop, 
which is not computably-orderable ? 

(Sourse: change "computable" to "Borel",  then 
2-adic numbers is an example.) 

V.Kanovei