Today I have fired up a tanning lamp which has a clear 125W high-pressure mercury burner in it. From what I saw and read on the net, I always thought that clear MV arcs are blue-green, but nope! It was brilliant cool white in color, maybe a bit blue-white but not green at all!

Maybe it's my eyes? How do you perceive the color of clear mercury bulbs?

I have only seen one clear merc with my own eyes. I would call the color green-blue. and ofcourse there were no red color at all. 

If it's a tanning lamp, you should NOT be looking at it directly! Hopefully you are viewing it behind glass. I had one of these in an old "Sunlamp" fixture - and like all clear mercury lamps, it is a brilliant white, but with a blue-green tinge, which is more noticeable when you either view the lamp from a distance or view its reflection and certainly in photographs.

Also, if you spend enough time under most any light, your eyes will adjust to it so it appears more white. I remember spending several hours in a purely HPS-lit gymnasium...it seemed normal after a while, and then incandescent lamps and daylight looked very strange afterwards.

BTW some of the mercury sunlamp lights are self-ballasted, so they will appear more white due to the blended light from the arc tube & incandescent filament. The latest HID tanning lamps are metal halide - I don't know the halide salts used in these, but they seem to appear "white" like ordinary MH lamps.

This one is ballasted by heater elements which do not give off any visible light.
And yes I was wearing sunglasses and didn't look at the burner directly.

I pointed it at the wall and when I walked through the house and then looked back into my room, it was almost like being lit with ~6000-7000K fluorescents, with absolutely no green tinge (here I just looked at the reflection on the wall without anything over my eyes).

What i have an experience, mainly with HR50W/S i sometimes use above my desk ( here ), clear (or not color corrected, as the my one is only diffused) lamps itself are looking blue-white (i think mix of blue with green/yellow is responsible for it), but objects they are illuminating tend to look greenish (wall, desk; the blue is relatively more attenuated, so the yellow/green peak out). Definitely the red is missing: What is normally red, look VERY dark... Special things are office papers, as they do contain some whiteners, what convert UVA to something (blue-green), so they appear unnaturally bright. And other thing is skin - as it normally absorb nearly all blue, the lamp gives it greenish-gray look. And all hematoms, even those normally invisible (assumed as healed), pop up, so the one's overall look is really scary, really like Zombie...