You could try nitrocellulose dope - flying model aircraft makers used to use it, and it is still widely available in cans at hobby shops. It makes cellulose rich fibres (which cardboard is) stiff and waterproof.
Now, I have teied taping Skywatcher boxes. I am not sure what they have in it, but most tapes won't adhere properly to it. I have been reliably told all the SW boxes are recycled, ex-USA paper. I remember having GREAT difficulty making tape stick to a SW box when I went to post it.
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