TL;DR: mixing warm white and cool white LEDs: not worth it

So here's a quick experiment that was kind of a failure. I document it here to save others the trouble of trying it.

First some background: different kinds of lights can render colors more or less faithfully. You may have noticed that some streetlights, like orange low-pressure sodium bulbs, make everything look orange: blue objects look black because there is no blue in the streetlight spectrum.

How well an illumination source can render colors is characterized with the Color Rendering Index (CRI), a number which tells you how close to "natural" your illumination source is, where "natural" is pretty close to sunlight with a CRI of 100. To render colors well, the source illumination should have a wide spectrum, that is, contain a mix of all different colors. Light sources with gaps in the spectrum, like some vapor discharge tubes and some phosphors, are missing colors that are necessary for good color rendering.

The actual details get somewhat technical, but a CRI of 100 is close to sunlight: halogens are pretty near that; fluorescent lights are in the 80s to 90s, anything below 80 is not great and colors will look wrong.

White LEDs have a mix of colors from a bright blue LED and a yellowish phosphor. Early white LEDs had pretty poor color rendering, you may remember their ghastly whitish-blue color, with little warmth in the spectrum. Recent advances in phosphors have improved LED CRI pretty dramatically: good ones are well into the 90s, and even inexpensive strip lighting can be pretty decent. Typical white LEDs available today have a "warm" color temperature with a yellow-orange phosphor or a "daylight" color with a yellow-green phosphor.

To get to the experiment: I wanted to see if combining a warm and cool LEDs would improve the color rendering of the combined light. So I obtained some LED strips of both kinds, and hybridized them by replacing alternate warm LEDs with cool (and vice-versa). Theoretically the combined spectrum should be wider than the individual LED types and so improve the CRI.

This was not that difficult but a little painstaking. The white plastic housing of these LEDs is thermoplastic and will melt under high heat, but I was able to desolder them fairly painlessly by holding the LED strip over a hot-air rework gun. (Surprisingly, the adhesive underneath was mostly intact, I was expecting a sticky burned mess.) Once I had desoldered 1/2 of the LEDs from each variety of strip, I took the loose ones and soldered them to the other, yielding two strips of alternating warm and cool LEDs.

RESULTS:

Nope. Though there's no easy way to objectively measure the CRI, to my eyes there was no detectable improvement. Moreover, the combined light had a slight greenish cast, making things look very faintly sickly. I will guess that the warm phosphors tend orange (reddish green -- no, really!) and the daylight phosphors are greenish yellow, so the combination had an extra dose of green making it a little unbalanced. But it's a very slight effect and may not even be noticeable unless you are looking for it. I still use a combined strip over my workbench -- but I can enthusiastically say the results are not worth the trouble!