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When two surfaces are exactly coplanar most renderers get confused on a pixel-by-pixel basis as to which one is 'in front.' This is a longstanding and known issue. However...

In Maxwell, when two surfaces are exactly coplanar but clipping masks fully hide at least one of the overlapping regions, the fully invisible portion of surface A can still cause errors in surface B.

Bug? Or simply a natural state of affairs? And can it be addressed?

(I don't need it addressed, but I can see a future use for someone for (as an example) 'floor tilings' that rely on well-behaved invisible tile portions)

(I'm doing something based on this old tapestry, btw...)

Do you specifially need that to be two objects though? Because I think it would be handled much better via material layers and one object instead.

I'm fine with just pushing one of the planes a little bit in Z. I just thought you might find it interesting that the invisible portion of one plane can interfere with the visible portion of another plane.