Applying lut to HDMI output

[kolpack]

I'd like to be able to use Sony cameras (preferably A7siii or A7C) for live broadcast, with color grading or lut applied to the HDMI output in real time (again, live broadcast). Is this possible? Sorry if this is a n00b question. I'm coming from Blackmagics where this is relatively straightforward....can't find info about this specifically for Sony cams. Thanks in advance!

[VTC]

Atomos Ninja V. Load LUTS. Send signal to Ninja from camera. Select LUT. Output live broadcast LUTted image. 

(Be advised your output image will be forever stained with whatever LUT you selected and why anyone would volunteer for this applied LUT, painting yourself into a corner type of disaster is behind my comprehension.)