I've found a website (Pearson's BioCoach) that claims centrioles duplicate in Prophase II. Is this accurate? Does it depend on the species in question?
Looking at three textbook illustrations of meiosis, none of their illustrations are consistent. All depict the movement and number of centrioles through Telophase I the same way: centrioles replicate in S of Interphase, then do their thing through Meiosis I. We end up at Telophase I with 2 centrosomes, each with 2 centrioles. There is one centrosome at either pole of the cell.
After that, the diagrams are unclear:
One text (general biology) seems to indicate either centrioles or entire centrosomes replicate after Telophase I, because Prophase II is shows both cells with 2 centrosomes, each with 2 centrioles.
Another (basic genetics) illustrates 2 daughter cells of Meiosis I inherit one centrosome each following Telophase I. In Prophase I, the centrioles in this centrosome splits, with one centriole going to either pole. Cytokinesis after Meiosis II results in 4 daughter cells, each with one centriole.
A third (more thorough genetics) stops depicting centrioles after Telophase I, instead showing the spindles growing in an aster formation from an empty space, the space where there were 2 centrioles up until Telophase I.