Character-based writing systems
In Taiwan today almost everyone has at least a basic standard of literacy in written Mandarin. This pre-existing character literacy is used by some as a platform for writing Taiwanese in characters.
For words in Taiwanese that are cognate with Mandarin words, this presents no problem. The Taiwanese word for dog (káu) and the Mandarin translation gǒu both share a common semantic root, and so there is no difficulty in using the same character (狗) for both.
However, other words don't have immediately obvious cognates in Mandarin; for example the word lóng, meaning 'all, completely', is often used where the word 全 (quán) would be used in Mandarin - but the two are not related. The two main methods of writing Taiwanese with characters differ in how they approach this problem.