Tried usually 3m clear double sided strips that interlock like Velcro, but weren’t strong enough.
How to hold plexiglass panels for an old storm door?
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Two large plexiglass panels over the screen inserts.
How about using a glue? A plastic adhesive should work
There are silcone products available that you can use a caulking gun to create a bead along the edge of the plexiglass plexiglass. “No Drafts”, DAP produacts. Check with your local hardware store.
The screen inserts should be removable for cold months. Put plexiglass in its place. Use mirror clips or screen clips if needed.
Agree with Rappunzel, that's what I would do.
I used clear silicone when I change to plexiglass after breaking the glass just about every time I would mow the lawn.