"thick veil" is correct and usable in written English.
You can use it to describe a barrier or divide between two things--for example, "a thick veil of secrecy" or "a thick veil of mistrust."
Like ADIA, ADIC operates behind a thick veil of secrecy.
ROME — The Vatican drew aside, for a brief and puzzling moment, the thick veil that covers its internal business.
A THICK veil of snow had settled on Kilimanjaro the morning after my group arrived in Tanzania.
That is, however, very hard to be sure of because they all operate behind a thick veil of secrecy.
Once the planets were destroyed, the debris would have fallen onto the star, generating a thick veil of dust and gas.
The life of Pythagoras and the origins of Pythagoreanism appear only dimly through a thick veil of legend and semihistorical tradition.
The female anchors are covered from head to toe, their faces hidden behind thick veils.
Thick veils of gas and dust shield the core from our view.
Years of burning stupendous amounts of coal have shrouded its cities in thick veils of smog, and the country has dramatically scaled back its coal use since consumption peaked in 2013.
Most of us are so wrapped up in thick veils of irony and projected cool that the line about Kierkegaard that Chris Kraus trots out in I Love Dick that suggests that, "the sign will always triumph through the screen of an ironic signifier" is about as true and accurate as me saying that I'm both incredibly wealthy and incredibly handsome.
Northern residents, on the other hand, seem to desire a thicker veil between themselves and New York City.
