Is the phase of plane waves a frame-independent quantity?

The invariance of the phase of plane waves among inertial frames is investigated in some details. The reason that eventually led the author of a recent EPL letter [EPL \textbf{79}, 1006 (2007)] to a spurious conclusion of the non-invariance of the phase of waves has been identified -- it is the ignorance of the effect of relativistically-induced optical anisotropy in the analysis of the problem. It is argued that the Lorentz-invariant expression for the phase of waves should be taken in the form $\Phi=\mathbf{k\cdot r}-\mathbf{k\cdot u}/c$, instead of the widely-used expression $\Phi=\mathbf{k\cdot r}-\omega t$ which has a limited validity.