How do I create a transparent stroke above other shapes?

Hello

I have the following shape:

What I'm trying to do is give the blue box a transparent stroke that will "cut through" the red boxes and show the background underneath, whatever that may be ...white, another color, or a photo.

I can easily achieve this when I give it a white stroke. It'll look like this:

When I change to a background color other than white or even a photo however, I end up with this problem:

How do I make it so that I have a transparent border (not white) around the blue box? Which effectively cuts a "transparent stroke" (space) between the parts where the blue and red boxes intersect?

Thank you very much for any advice I've been trying to get this done for days without success, so you're my last hope!