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The question might be addressed by considering a specific subfield:

 x=QQbar(7**(1/8))
 L.<x>=NumberField(x.minpoly())
 Ls=L.subfields(degree=2)           
 print(Ls)
 G=Ls[0][0].galois_group(names='x')
 print(G.list())

In this case, the subfield which is real is of degree 2, seen as a vector space. The number field can be restricted to the reals by considering the appropriate subfield which is a proper subset of the reals. The output of the above code is

[
(Number Field in x0 with defining polynomial x^2 - 7, Ring morphism:
  From: Number Field in x0 with defining polynomial x^2 - 7
  To:   Number Field in x with defining polynomial x^8 - 7
  Defn: x0 |--> -x^4, None)
]
[(), (1,2)]