code: def real_rotated_Gammas(m,l,phi1,phi2,theta1,theta2,gamma_ml):
      
      
      if m>0:
          ans=(1./sqrt(2))*(rotated_Gamma_ml(m,l,phi1,phi2,theta1,theta2,gamma_ml) + \
                            (-1)**m*rotated_Gamma_ml(-m,l,phi1,phi2,theta1,theta2,gamma_ml))
          return ans.real
      if m==0:
          return rotated_Gamma_ml(0,l,phi1,phi2,theta1,theta2,gamma_ml).real
      if m<0:
          ans=(1./sqrt(2)/complex(0.,1))*(rotated_Gamma_ml(-m,l,phi1,phi2,theta1,theta2,gamma_ml) - \
                                          (-1)**m*rotated_Gamma_ml(m,l,phi1,phi2,theta1,theta2,gamma_ml))
          return ans.real
docs: This function returns the real-valued form of the Overlap Reduction Functions,
      see Eqs 47 in Mingarelli et al, 2013.