I know exactly what it is and have had it on many occasions. First off, we need to understand how lye is made:
*Lye was (and is) commonly made by perforating the bottom of a large pail or bucket
*Now line the bottom with a few layers of cloth
*Next, fill it with the white ash of clean burnt wood
*Pour in water that would work its way down through the bottom of the bucket
*The white liquid is your lye. Use it as a disinfectant or boil it down with tallow and salt to make lye soap.
I mention this because mixing ash with water or food for purification was a common practice dating back for centuries. According to what I've found so far, Whites, namely used it to decontaminate water but picked up using it in cooking through their contact with Native Americans:
"They (the Cherokee and Creek) use a strong lixivium prepared from ashes of bean stalks and other vegetables in all their food prepared from corn, which otherwise, they say, breeds worms in their stomachs."
-William Bartran, Transactions of the American Ethnological Society Vol. 3, Pt. 1 (1789)
"Water taken from stagnant pools, charged with putrid vegetable matter and animalcumulae, would be very likely to generate fevers and dysinteries if taken into the stomach without purification. It should therefore be thoroughly boiled and all the scum removed from the surface as it rises; this clarifies it, and by mixing powdered charcoal with it, the disinfecting process is completed."
-Randolph Marcy, The Prairie Traveller p. 49 (1859)
This is still practiced, BTW. I always mix it into my cornbread and sagamite. It is also used in other traditional Native American bean dishes & bean breads. It imparts a mild effect on the flavor that is actually good. When I dig for my water, I allow the pool to settle; boil the water thoroughly; and then sift in about a 1/2 tsp of white wood ash per quart.
The soldier is referring to purified water taken from a standing source as opposed to a well that is less prone to contamination. In other words, Alva Greist was complaining how his overall breakfast experience sucked; bad weather, wet kindling, trouble with the fire, mundane food and coffee from a 2nd rate water source that had to be cleaned before consumption.
-Dave
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