Learn English – Why do “catsup” and “ketchup” coexist

John Ayto, Arcade Dictionary of Word Origins (1990) has an interesting entry for ketchup that agrees in part with Etymology Online's analysis (cited in Unreason's answer):

ketchup {17 [century]} Ketchup is a Chinese word in origin. In the Amoy dialect of southeastern China, kôechiap means "brine of fish." It was acquired by English, probably via Malay kichap, towards the end of the 17th century, when it was usually spelled catchup (the New Dictionary of the Canting Crew 1690 defines it as 'a high East-India Sauce'). Shortly afterwards the spelling catsup came into vogue (Jonathan Swift is the first on record as using it, in 1730), and it remains the main form in American English. But in Britain ketchup has gradually established itself since the early 18th century.

Glynnis Chantrell, The Oxford Dictionary of Word Histories (2002) tells a different story:

ketchup {late 17th century} This is perhaps from Chinese (Cantonese dialect) k'ē chap 'tomato juice'.

Bryan Garner, Garner's Modern American Usage, second edition (2003) weighs in with this assessment:

ketchup, catsup, catchup. The first spelling greatly predominates in modern usage. It has the advantages of phonetically approximating and of most closely resembling the word's probable source, either the Cantonese k'ē chap or the Malay kēchap, both referring to a kind of "fish sauce." The pronunciation is either kech-əp/ or kach-əp/; kat-səp/ is pretentious.

Robert Hendrickson, The QPB Encyclopedia of Word and Phrase Origins, fourth edition (2008) has this lengthy discussion:

ketchup. Is it ketchup, catsup, catchup, or kitchup? Since the word derives from the Chinese Amoy dialect ke-tsiap, "pickled fish-brine or sauce," which became the Malay kechap, the first spelling is perhaps the best. The original condiment that Dutch traders imported from the Orient appears to have been either a fish sauce similar to the Roman garum or a sauce made from special mushrooms salted for preservation. Englishmen added a "t" to the Malay word, changed the "a" to "u" and began making ketchup themselves, using ingredients like mushrooms, walnuts, cucumbers, and oysters. It wasn't until American seamen added tomatoes from Mexico or the Spanish West Indies that tomato ketchup was born. But the spelling and pronunciation "catsup" have strong literary precedents, as witness Dean Swift's "And for our home-bred British cheer,/ Botargo {fish roe relish}, catsup and cabiar {caviar}." (1730). Catchup has an earlier citation (1690) than either of the other spellings, predating ketchup by some 20 years. Ketjap, the Dutch word for the sauce, and kitchup have also been used in English.

Joseph Shipley, Dictionary of Word Origins (1945) has a brief but interesting treatment as well:

ketchup. Sometimes spelled catsup, this word has no relation to milk; it is an oriental word: Malay kechap; Chin. ketsiap, Jap. kitjap; meaning a sauce, as the brine of pickled fish. Our most familiar form is tomato ketchup.

Perhaps the most striking thing about Shipley's entry for ketchup is his spelling of "our most familiar form" as "tomato ketchup." Shipley was an American writing at the end of World War II. If you check the Ngram chart below, you'll see that catsup was substantially more common than ketchup in Google Books content published in 1945, and had been for most of the previous three decades.

Ernest Weekley, An Etymological Dictionary of Modern English (1921) favors the spelling ketchup even more decisively, referring to catsup as an example of "folk-etym[ology] perversion." Weekley's entry for ketchup is bare-bones:

ketchup. Malay kĕchap, ? from Chin. ke-tsiap, brine of pickled fish. With incorr[ect] catsup cf. Welsh rarebit.

It is certainly true that what the English understood by ketchup was a spicy sauce dominated by fish, as is evident from the recipe for ketchup that appears in Charles Carter, The London and Country Cook: Or, Accomplished Housewife (1749): which specifies using "twelve or fourteen anchovies" with less than a pint and a half of wine vinegar and port, plus shallots, horseradish, ginger, pepper, mace, nutmeg, and lemon peel.

The Ngram chart for ketchup (blue line) versus catsup (red line) for the period 1700–2005 is volatile:

Overall, ketchup broke away from catsup only in the early 1980s—a time frame that roughly coincides with the shift in spelling of at least two major brands of tomato ketchup from catsup to ketchup. According Aisha Harris, "Is There a Difference Between Ketchup and Catsup?" in Slate (April 22, 2013), Del Monte switched its spelling to ketchup in 1988, and Hunt's did so "significantly earlier." But the same article reports that Heinz, the biggest U.S. purveyor of the stuff, originally sold the product "as 'Heinz Tomato Catsup,' but changed the spelling early on to distinguish it from competitors."

I suspect that the radically different trajectories of the two spellings since around 1980 are largely due to changes in product spelling by major purveyors of tomato ketchup during that period. That is to say, I can't think of any other circumstance in the past 36 years that would explain the change.