Must the characteristic impedance of a transmission line be non-real for it to be lossy?
Phil, the real part of the propagation constant is the attenuation constant and this equals: -

\$Re\sqrt{(R+jwL)(G+jwC)}\$ and not the formula you have in your question. 

The formula you have used is for characteristic impedance.

This wiki page should confirm this (right at the bottom): -



So, if you do the math at low frequencies (to make life easier) you see that the attenuation constant becomes \$\sqrt{RG}\$ and if R=G=1 then you have a constant of 1 and a lousy highly lossy line. A lossless line has a re(propagation constant) of zero.