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I know that Uncle Tom's Cabin is full of neologisms, and I try my best to grit my teeth and infer as best I can without racing down every such rabbit hole that presents itself, but with rhumatis, for whatever reason, I can't quell my curiosity. I must figure out what it means (if anything). The passage for context, from Chapter 10:

"Now," said Aunt Chloe, bustling about after breakfast, "I must put up
yer clothes. Jest like as not, he'll take 'em all away. I know thar
ways—mean as dirt, they is! Wal, now, yer flannels for rhumatis is in
this corner; so be careful, 'cause there won't nobody make ye no more.

Question

What is rhumatis supposed to mean here? Given that flannels are usually warm, I'm curious if its something to do with the outdoors. Or perhaps its some kind of phonetic neologism? Can't think of any words that sound anything like rhumatis though.

Rhumatis is almost certainly a colloquialism for rheumatism.

In the era that Uncle Tom's Cabin was written, rheumatism was a catch-all term for what modern medicine recognizes as distinct conditions and disorders of the joints and muscles. It is no longer in professional or academic use because, like ague, grippe, catarrh, and so on, the causes and treatments for different manifestations of of "rheumatism" vary, and must be distinguished clinically.

Second, cold, wet weather has long been associated with arthritis pain. Whether the connection has scientific validity is a matter of dispute (some studies ascribe it to barometric pressure), but this is irrelevant, as Aunt Chloe would have been following customary beliefs and folk medicine. The substandard living and working conditions of slaves would lend themselves to choosing a durable and insulating fabric for clothing or blankets to ward off the effects of weather on one's rheumatism or, in the case of the children, perhaps help to prevent it.

Greek rheumatismos, coined by Galen of Pergamum, a philosopher, physician, and pioneer of medical practice, in the 2nd century CE. Today, few if any doctors use the word “rheumatism” to describe a specific medical condition, although it still exists as a colloquialism.

Source The Rheumatologist

The 'rhumatis' spelling variant of 'rheumatiz', far from being a neologism, is fairly well-represented in fiction during the early decades of the 19th century, before the 1852 publication of Uncle Tom's Cabin. It appears, for example, in 1807 (col. 1, toward bottom), 1833 ("Temperance Anecdote"), 1835 (about mid-page), 1836 (p. 62, about the middle of the first para.; p. 109, toward bottom), 1838 (third paragraph), 1839 (col. 3, third para.), 1839 again (last para.; republished in book form in 1845), 1844 (next to the last para.), 1844 again and again. The variant also appears in an 1846 spelling book, where it rates a mention as "painful to the ear" (footnote).

OED (paywalled) reports two senses of 'rheumatiz'. The first sense, possibly attested from earlier than 1490, was certainly obsolete by the 1800s:

1. A disease caused by the abnormal production or flow of rheum. Obsolete. rare.

Sense 2, attested from 1760 to 2008 in OED, equates the meaning of 'rheumatiz' with the meaning of 'rheumatism', and states that the word is "[c]hiefly British and U.S. regional." The report that the form 'rhumatis' is "British and US regional" jives with the uses I turned up in the first half of the 1800s, although solely on the basis of those uses I would have been more inclined to honor the Scots.