As long as all of the light sources are close to ideal black body radiators, then the spectrum of light they emit will be "neutral" with respect to plus/minus green, and shouldn't contain aberrant spectral spikes. Sunlight and tungsten light are both close ideal; black body radiation at a temperature of 5600ºK is very close to the color of sunlight, and 3200ºK is close to the color of tungsten light.
If you mix the two in your scene, then you can arbitrarily choose what's "white" by moving your software's white balance slider between 3200ºK and 5600ºK. When you push the slider towards higher numbers, the places in your scene which received tungsten light will appear orange, and the places with daylight will appear white. When you push the value lower, the places with tungsten light will appear white, and the daylight-lit places will appear blue. Anywhere in between, and you'll get a mixture of orange, neutral, and blue. The value you ultimately select is a completely subjective, artistic choice. There is no right or wrong.
Fluorescent lights and LEDs (especially cheap ones) contain spectral spikes, and aren't close to ideal black body radiators. If you light a grey card, white card, or color chip chart using a poor source (low CRI), and then set your software's WB to that color (pick/eyedropper the card), then all of the natural, pure white light sources in your scene will look strange.
There's nothing inherently wrong with mixing the color temperatures of your lighting as long as you understand what you're doing, and what effect it will have on your image. If you do it well, it can give your images extra color contrast and depth.
On the other hand, mixing color temperatures of light in your scene effectively reduces the dynamic range of your camera because one color channel will clip before the others. When you set your camera's white balance, you're optimizing the camera to receive that color of light. If you set your white balance for tungsten, and some bright sunlight creeps into your scene, the blue channel will clip disproportionally from the others, causing ugly, mis-colored, and irreparable hot spots, for example. The inverse is true for tungsten, of course. But as long as you expose carefully enough that none of the color channels clip, this shouldn't be a problem.
With that said, if you want to make the most out of a limited bit depth, the best thing you can do is avoid mixed-temperature lighting, and set your white balance as accurately as possible in camera. The image should appear "white" or neutral. Then, if you want a cooler or warmer look, adjust the color temperature in software. This maximizes the use of your camera's dynamic range, and still gives you artistic control over your image's overall temperature.
Your brain adjusts to whatever the dominant source of light in a scene is, and it does it very quickly. If you only have one source of light in a room, you'll perceive a grey card to be neutrally colored. It's only when you have external daylight creeping in through the periphery that you'd perceive the card as warm. Your visual perceptive system is very adaptive, which can mislead your intuition. This is why colorists rely on vectorscopes and waveform monitors to objectively judge what they see. Cameras are much less flexible than the human vision system, which is why it's often surprising when a scene doesn't fit within the camera's dynamic range; overexposed, underexposed, or off-balanced.