Previous studies have shown that multilingual speakers are influenced by their native (L1) and non-native (L2) grammars when learning a new language. But, so far, these studies have mostly used untimed metalinguistic tasks. Here we examine whether multilinguals' prior grammars also affect their sensitivity to morphosyntactic constraints during processing. We use speeded judgment and self-paced reading tasks to examine the comprehension of German possessive pronouns. To investigate whether native and non-native grammars differentially affect participants' performance, we compare two groups of non-native German speakers with inverse L1-L2 distributions: a group with L1 Spanish -- L2 English, and a group with L1 English -- L2 Spanish. We show that the reading profiles of both groups are modulated by their L1 grammar, with L2 proficiency selectively affecting participants' judgment accuracy but not their reading times. We propose that reading comprehension is mainly influenced by multilinguals' native grammar, but that knowledge of an L2 grammar can further increase sensitivity to morphosyntactic violations in an additional language.