One of the major aims of discourse-processing literature is to understand whether and when readers form discourse-level representations online. To test this, two word-by-word, self-paced reading experiments investigated the time course of integrating incoming information about the protagonist into the unfolding discourse-level representation in Tunisian Arabic (L1) and the role played by task demands in such integration. Discourse-level anomalies related to character-attribute information of the protagonist were examined under the comprehension instruction alone (Experiment 1) and under the condition of adopting-the-protagonist-perspective instruction (Experiment 2). The present experiments revealed two major results: (1) Global coherence is not maintained online in the absence of specific goals in L1 text processing; that is, readers integrate incoming information into the unfolding discourse representation online when engaged in deep processing, which shows that task demands modulate L1 text processing. (2) Such online integration is delayed until wrap-up positions are formed. This resulted in a revisited model defining what might modulate the occurrence of online integrative discourse processes.