I've been perusing some "prepper sites," looking for ideas about boondocking with minimal pretty much everything. Backpacking experience should be super helpful there.
So,some preppers list foods that don't absolutely require refrigeration. Mayo is among them,with a few caveats. As others point out, introduced foreign particles can spoil. Preppers say to use a clean utensil to remove the mayo from the jar. For my purposes, purchasing in small containers should be helpful, as may be squeeze containers. If I'm on a trail, I do think packets are the way to go.