Potts (2005) provides an elegant and influential approach to the syntax and semantics of a wide range of 'parenthetical' constructions, including various kinds of supplementals/appositives and expressives, which have not received much attention in the LFG literature. The prospect of wholesale importation of Pottsian analyses for such constructions is appealing. However, there are (as Potts himself notes) problems with his approach from a resource sensitive perspective, such as is often assumed in LFG approaches to the syntax semantics interface. Potts sketches, but does not develop, a way of overcoming these problems. In this paper we attempt to develop his proposal and explore ways in which it can be implemented in the architecture of LFG. For exemplification, we focus on supplemental ('appositive', non-restrictive) relative clauses in English. A side effect of our discussion is thus a proposal for an LFG analysis of this construction, which appears to have been neglected hitherto.