Poko (Skou, Papua New Guinea) displays a complex tone system, with three contrastive levels, toneless syllables and floating tones. Lexical tone is characterised by robust patterns of anti-alignment, wherein high (H) tones may not be initial and low (L) tones may not be final (McPherson & Dryer 2021; McPherson 2022). This article analyses the postlexical realisation of tone, especially the behaviour of floating tones, rising tones and toneless syllables. The Poko tone system shows unique twists on cross-linguistic patterns, such as the OCP, tone raising before L and the avoidance of non-final rising tones. We demonstrate that the behaviour of floating tones cannot be accounted for in a constraint-based theory with global evaluation, as in traditional Optimality Theory or Harmonic Grammar. Instead, the data motivate the use of directional Harmonic Serialism (Lamont 2022b), wherein changes are made incrementally and directionally, thus avoiding the creation of ties among otherwise similar candidates.