Thanks - yes, I tried that. Using a png file with transparent colour did not work either.
I seem to remember that a Ricoh production printer I used some years back at the office did not show this problem.
I printed the Word file itself and a pdf file of it. Both gave this result.
Is there perhaps a printing option that avoids this?

I tried to create a replacement graphic, consisting of the same grey background. I then placed the irregular shapes on top. The exported image was ok, but after inserting it into the document instead of the table, the grey had changed.

So this path around the problem does not seem to work either.