Fluorescent can alter the enter behavior of color, including gray balance and this is not something the human eye will adapt to. A color temperature meter is insufficient. D50 is the standard. 5000K and D50 aren't the same thing. A spectroradiometer is what you'd need to evaluate the suitability of a light source, and compare its spectral power distribution to that of D50.
Spectral distribution is what's important.
Color temperature is totally misleading. I'm therefore posting three messages he sent me. Just for the curious, the spikes are due to the fact that fluorescent bulbs are based in mercury emissions energizing the phosphor coatings on the inside of the bulb which fluoresces at differing wavelengths. So fluorescent bulbs will always have some level of spikes due to their dependence upon mercury as a source.