Mach Zehnder phase change?
Warning: This answer is based primarily on university courses on optical networks. There could be a bit of real-life missing.

So in traditional single-wavelength systems, we have usually considered the assumption that \$V_{\pi}\$, the half-wavelength voltage, is constant for each optical band, but that the value between the bands will usually be different, to be good enough. In traditional systems, the 1550 nm band is quite popular, so quite often, there's not much need to run multi-band systems. 

In reality, the \$V_{\pi}\$ is wavelength-dependent even within the same band, but the differences are usually small. So what happens in real-life is that you'll set the voltage to some value and that value will cause different phase-shift for each wavelength going through the modulator. This will give you a limit on the usable bandwidth of the signal going through the modulator. For example, Analysis and Improvement of Mach–Zehnder
Modulator Linearity Performance for Chirped
and Tunable Optical Carriers paper gives some bandwidth values for different types of MZMs. In any case, the bandwidths are in the terahertz range, so for "normal" applications, this is not a big issue, but for very wide DWDM, this could be problematic.

In the end, it's the duty of the engineer designing the system to determine what's good enough and what could be problematic.