Shipping/storing unvarnihed egg tempera paintings

With unvarnished, high PVC paints like traditional egg tempera the critical thing is to chose something that causes the least abrasion and then cover this with something that offers physical protection. I personally dislike glassine for almost anything. It tears too easily, is rather sharp when folded or creased, and easily sticks to varnished/oil paint surfaces (not an issue here). The gold standard is silicone release mylar (coated on only 1 side) with the silicone towards the painting surface allowing for taping in the non-silicone side. Facing this will double walled cardboard, or even better a rigid panel, would be great, although surrounding the SRM with bubble wrap is suitable. This is perfect for oil and acrylic paintings or any varnished artwork but is probably this is overkill for you.
I remember when working as an art handler, pre-conservation grad school, my boss used a very soft for of Tyvek for a similar purpose and it seemed to work very well. I believe that it was Tyvek Soft Structure (Grade 14-M) but unfortunately, before I had the foresight to ask about the particulars, my boss sadly became sick, retired, and passed away from thyroid cancer.
I hope others here can speak on the subject.

I like Brian’s solution.  Mine is low tech but so far has worked for me.  For framed work:  I lay the frame, face down, on thick, rigid cardboard (for short term coverage, i.e. when shipping work) or thick, acid-free matboard (for long term storage), then use a pencil to outline the frame on the board. I lift up the frame, then cut the board with a metal ruler and x-acto knife, to end up with a piece of board the exact size as the frame.  I place the board carefully over the frame, which leaves air space between the painting and the board; then I wrap the whole thing in kraft paper, like a present – very tightly, so that the board can’t shift – and tape the kraft paper securely.  I wrap all of that in bubble wrap.  For unframed work, the process is similar – I cut a piece of paper (plain copy paper for short term; acid free paper for long term) and a piece of cardboard or mat board the exact size of the painting panel; place first the paper, then the rigid board, on top of the painting; warp the whole thing in kraft paper, like a present, and tape securely; then warp it all in bubble wrap.   By the way, do not put bubble wrap directly on a tempera painting – I’ve seen it leave a bubble wrap impression on the surface.

Due to high PVC, the surface of an egg tempera painting is (on a microscopic level) irregular, porous, open and absorbent. This makes it vulnerable to moisture entering in (its high PVC ground can also attract moisture); and dust/dirt embedding within the paint surface.  It’s also vulnerable to abrasion, and animals/insects nibbling at the paint. My solution addresses most of these issues, but not the moisture problem – a covering made from a synthetic material (as Brian recommends) would protect the front from moisture better than paper or board, but still the edges and back of a painting may let in humidity (which is why it’s recommended to paint edges and back with latex or alkyd house paint).  Egg temperas – whether hung on a wall, in transit or in storage – are especially vulnerable to moisture, and too much relative humidity can cause egg temperas to grow mold, flake, delaminate (I’ve seen all three).   So in addition to how you wrap a work, be sure to keep it in a not overly humid environment (and/or in an environment with good air circulation).