This paper argues that the analysis of Japanese internally headed relatives must steer a course between the Scylla and Charybdis of reduction to externally headed relatives and reduction to discourse anaphora. The analysis given here is based on that of Grosu & Landman (2012), but the present paper reconsiders some of the central data in Grosu & Landman (2012) in the light of the diagnostic tests given in Grosu & Hoshi (2016), and argues for a simplification of the basic analysis, which actually strengthens the theory considerably. The paper then extends the analysis given with an analytic tool – a Lombardian presupposition mechanism for Kuroda functions – which allows the implementation of the Kuroda relevancy condition (Kuroda 1976–77). It is further shown how the improved analysis can provide a semantics for so-called change relatives, which were mentioned but not analyzed in Hoshi (1995) and Kim (2007). An appendix recapitulates and refines the discussion of scope phenomena in internally headed relatives from Grosu & Landman (2012).