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The cheapest that I can find is given in UK sterling (omegaflightstore.com) but you could look for the same flights elsewhere. It's a heck of a long trip though with long stays in Taipei. It's about A$1820 with China Airlines.
Is price the most important criteria?
How about...
29/6 Virgin Australia + Etihad Airways 11:00 BNE-xSYD-xAUH – 07:20 LHR (29h 20m)
9/7 14:45 LHR-xAUH-xSYD – 09:30 BNE (33h 45m)
with Expedia $2445
plus some other options using these airlines
Etihad for about 2270 booked through their own website - probably the same flights as the Virgin booking but they're usually cheaper than Virgin and don't charge credit card booking fees. Fairly direct too. You don't get a lot quicker than 27 hours.
Just don't book too soon. I'm flying Sydney to Dublin on Friday 7th June - watched the fares for the exact same dates for weeks at about 2200, to suddenly see them increase to mid- to high-2000s. Imagine my cursing. But a few days later they came back down to about $1850! Snapped it up.
Some people say you should clear your computer's internet browser cache (cookies etc) every so often, so the websites don't remember you from your last visit, so they can't play tricks with you (like lying about availability of particular fares).
Fares jump around all the time, so just spend a few minutes every day checking them. Try from different computers too (like at work, or a laptop or a library) to see if you get different results.
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