Oca plants start off fine and upstanding, but then they always over-extend themselves and flop over. The newly horizontal stems then send out secondary shoots, upright at first, but they too collapse if they get long enough.
Once day-length reduces to a certain critical point, new side shoots start to stolonate, heading downwards and forming tubers, at or just above ground level.
In Tubers - Big and Few or Small and Many? I made the observation that these tubers never reach the size of those that form wholey underground around the original planting tuber, and I speculated that preventing these stem tubers from forming might lead to bigger underground tubers.
I've since looked at photos of traditional Andean cultural methods. Oca is earthed up, but the ridges are much bigger than those commonly used for potatoes - perhaps two feet high - so big that the stems are held fairly upright even when the plants are large (have a look). Perhaps this is the main benefit of earthing up - rather that the supposed frost protection effect, or to 'encourage stolon formation'.
I'm guessing that the above-ground tubers won't form if the stems do not sense soil nearby. The plant's reserves will have to be relayed to the underground tubers, increasing their size, instead of being dispersed across all of those tiddlers that mostly end up as bird food.
As a simple test, I'll try a few plants growing up through a wire mesh cage. This should be enough to support the stems and maybe keep them from stolonating.
Let's see how it goes.
Here's the view on the 4th July,
...and on the 1st of September.
On the 21st October there was an unusually early frost, killing off other oca plantings, but this group escaped fatally serious damage, presumably by being above the coldest air.
It produced a reasonable crop (interestingly, with almost no tuber stems) but I had no other plants left to compare against.
If nothing else, this method allows Oca to be grown in a much smaller space than if they are allowed to sprawl.