As a permanent resident, you should. As a mere foreign spouse, you don’t.
There’s an inconsistency between Art. 52 of the Employment Service Act and Art. 9 of the Labor Standards Act, so even though an employment contract can be for an indefinite (permanent) term, a foreigner’s employment contract cannot, unless the foreigner is subject to ESA Art. 51 Par. 1 (including permanent residents). Mr. American Teacher’s contract had no visible expiration date, and he was married to a Taiwanese, but nonetheless he was not a permanent resident and not subject to Art. 51 at the time of the disputed resignation, so the contract was deemed a fixed term contract in accordance with Art. 52. Presumably, he argued that this is inconsistent with the Treaty, yet the Supreme Court merely stated that Art. 6 (not Art. 2) of the Treaty is inapplicable because it concerns due process, protection of life and property, and so on, not employment rights.