Today we think we figured out the culprit: I've inadvertently trained myself to not be able to endurance run.
Basically, my exercise experience in the past has been martial arts, and sprinting. These both have something in common: when you need to move fast, you have to get up on the balls of your feet. When you need to brace/stop/otherwise make sure you DO NOT MOVE, you drop your heels. This is trained into you until it's complete subconscious instinct.
2 + 2 = D'OH!
So yeah, today Fox noticed that apparently my body's instinctive reaction to my conscious brain saying "OK, jog" is to take the leg/foot position and stride length of sprinting and slow it down to about 1/5 speed. As you can imagine, this is inefficient as hell and drains your energy like whoa, awkward & straining to the muscles and tendons, and treats your internal organs like you're riding a trotting pony on top of a washing machine during spin cycle. No wonder I've been horfing.
But now that we know the problem, Fox and I can both work on making sure that I actually land on my heels and take long running strides instead of angled sprint-bites. Hopefully this will appease both the tendon and my stomach in the near future.