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is it possible in gimp to produce a display with a transparency that shows whatever is on the computer screen rather than the grey squares in the parts that are "transparent"
If you consider that Gimp is not really a content delivery system but more a content creation system, then the reason why this doesn't happen is obvious. Exporting your animation as GIF preserves the transparent background. From there you can put the image onto another background or import it to a new image in Gimp.
I think you are trying to achieve a video editing effect where your transparent background animation is used as a layer in a video timeline.