Yes. Here's a sketched example: Start in L. Let P be the forcing which adds ω<sub>1</sub> many Cohen reals, and let G be an L-generic filter for P. Then L(ℝ)<sup>L[G]</sup> will model ZF, but will have no well ordering of the reals. The point is that if σ is an automorphism of P, then σ can be extended to an elementary map from L[G] to L[σ[G]], and this extension will fix L(ℝ)<sup>L[G]</sup>. So if there was a well ordering of ℝ in L(ℝ)<sup>L[G]</sup>, it would give a well ordering of G which was fixed by σ. But σ can reorder the elements of G because of the homogeneity of P.