This word manifests a schism between spelling and pronunciation – and between two ways of pronouncing it. One might say there is a (somewhat repaired) historical schism in its form, too, but really it’s a manifestation of language change over centuries… followed by atavistic orthographic reversion, which in turn has given rise to a recrudescent pronunciation. Even more amusing, the version preferred by prescriptivists is a medial, not original, form – and it goes against the spelling, by contrast with such reverted spelling pronunciations as for the suffix ing, which was [In] for a long time until brought back to match the spelling by the yardstick-wielders.
But perhaps the thing to establish first is that this word does not sound like schist with an m swapped for the i, even though both words come from the same source: Greek schizein, verb, “split” (you will recognize it in schizophrenia, “split head” by origin). Instead, this word – which has been used in English since the 14th century, in reference first to ecclesiastical splits and then, starting a century later, to splits in general – passed into English already mutated as cisme or scisme, with [s] at the beginning. Then, with interest in and awareness of the classics growing, in the 16th century it got changed to schism to match the Greek (and intermediary Latin) source. And only over time, and with considerable resistance, and only by some speakers, has the onset come to be said [sk].
So let us look at the two pronunciations severally. The first presents us with a silent ch, which I defy you to find in another English word. It hisses and sizzles, and snips like scissors – which word, by the way, is unrelated. The vowel can almost be subsumed into the fricative flow: see how slightly you can separate your tongue from your alveolar ridge, so that rather than cleaving, it cleaves. Then decide whether to ply it slightly again before the [m] or dive straight in without even a hint of a schwa. One syllable or two? Or is this a word that may be used in arguments against syllables as an essential unit, or at least in favour of fractional syllables? In any event, though ism is a common suffix, few unsuffixed words (I was going to say “one-syllable,” but let’s not take that issue as decided) end with it: prism is perhaps the most common, and jism the least polite.
But if you wish to say the ch, you start the word with [sk], and thus enforce a schism between the fricatives. Certainly [sk] is common enough, and schism is not the only word that at first in English had a c (or s) pronounced [s] that then was respelled sch on historical grounds and is now sometimes pronounced with [sk]: schedule is the other one I can think of, and it has a similar history. But let me not get off schedule; I’m running long as it is. The skipping jaunt or sketchy scratch of [sk] lends this pronunciation a distinctly different air from the softer [s] version, a bit of a kick.
On paper, of course, it’s all one word, glaring at you and daring you to decide which way it will be pronounced, a sort of Schroedinger’s cat that could be either version or neither version until you ask which. And on the page it sometimes hangs out near religious and church, but its closest friend is between.