Much research has explored the degree to which not-at-issue content is interpreted independently of at-issue content, or the main assertion of a sentence (AnderBois, Brasoveanu, & Henderson, 2011; Harris & Potts, 2009; Potts, 2005; Schlenker, 2010; Tonhauser, 2011; a.o.). Building on this work, psycholinguistic research has explored the hypothesis that not-at-issue content, such as appositive relative clauses, is treated distinctly from at-issue content in online processing (Dillon, Clifton, & Frazier, 2014; Syrett & Koev, 2015). In the present paper, we explore the way in which appositive relative clauses interact with their host sentences in the course of incremental sentence comprehension. In an offline acceptability judgment, we find that appositive relative clauses contribute significantly less processing difficulty when they intervene between a filler and its gap than do superficially similar restrictive relative clauses. Results from two eye-tracking-while-reading studies suggests that recently processed restrictive relative clauses interfere to a greater degree with processes of integrating the filler at its gap site than do appositive relative clauses. Our findings suggest that the degree of interference observed during sentence processing may depend on the discourse status of potentially interfering constituents. We propose that this arises because the syntactic form of not-at-issue content is rendered relatively unavailable once it has been processed..