Yes, sarsaparilla is a real soft drink. According to my mother, it's correctly pronounced "SASS-pa-rilla". (She was a Phi Beta Kappa English major at UC Berkeley, so she was usually pretty trustworthy about pronunciation, but she may have been drawing on her Oklahoma upbringing for that one.)
I am quite sure that at least sarsaparilla is reasonably well known and available in Australia, and given that, probably in the UK as well. It's a bit old-fashioned, though. Probably better known by the oldies among us.
I can assure you that it's not just the okie accent, but a broader result of the Old West.
Being that it's origins are Spanish (
zarzaparrilla, lit. 'small vine bush') and it was popular in the Old West, it's probably a safe bet that the way we say it is a bastardization of the original pronunciation. Much like 'buckaroo' is derived from 'vaquero', or how Coloradans say 'La Junta' as 'luh-huhnuh' (yes, that's exactly how they say the name of the town). Oklahoma has Nah-VAY-duh (and a few others I can't remember offhand). Texas has a mix of bastardized and non-bastardized town names. Even Kansas has Sah-LIE-nuh (Salina) and El Doh-ray-doh (El Dorado).
Nebraska and Colorado are easily the worst about butchering foreign word city names, though.
Here is one popular Australian version.
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You may well be right about birch beer, though. I've not heard of that one.
I'm familiar with their ginger beer thanks to the World Market stores, but I didn't know they made sarsaparilla, too.
Birch beer has sort of a minty taste to it. Somewhat akin to Certs breath-mints, or a milder version of Altoids. Somewhere between the two. I'd recommend Sioux City brand over Frostop. Though, if Sprechers Gourmet Soda makes it, try theirs. They use honey as the sweetener in all of their recipes.
EDIT: I feel like an arse now. So here's the dictionary vowel symbol version for all of the international readers. Lŭh-hŭhnŭh, either 'Nĕvādă' or 'Nəvādă', 'Sălīnă' or 'Sălīnŭh' (I've heard it said both ways), and Ĕl Dō-rā-dō.