This question is similar to How many different vowels are there? but with a different twist: here I am not interested in the minimal difference between two vowels that can be heard, but in reliable and reproducable annotatability, meaning that the number of potential vowels is reduced by this requirement.
From anecdotal evidence, I have two data points: A scheme using only the cardinal vowels can be used with good inter-annotator agreement. On the other hand, Theutonista is too fine grained (it essentially allows for 25 shades of each cardinal vowel) and even the same annotator cannot annotate the same sample consistently when he re-annotates it some weeks later.