Different theoretical accounts of second language (L2) acquisition differ with respect to whether or not advanced learners are predicted to show native-like processing for features not instantiated in the native language (L1). We examined how native speakers of English, a language with number but not gender agreement, process number and gender agreement in Spanish. We compare agreement within a determiner phrase ("órgano muy complejo" "[[subscript DP] organ[subscript-MASC-SG] very complex[subscript-MASC-SG]]") and across a verb phrase ("cuadro es auténtico" "painting[subscript-MASC-SG] [[subscript VP] is authentic[subscript-MASC-SG]]") in order to investigate whether native-like processing is limited to local domains (e.g. within the phrase), in line with Clahsen and Felser (2006). We also examine whether morphological differences in how the L1 and L2 realize a shared feature impact processing by comparing number agreement between nouns and adjectives, where only Spanish instantiates agreement, and between demonstratives and nouns, where English also instantiates agreement. Similar to Spanish natives, advanced learners showed a P600 for both number and gender violations overall, in line with the Full Transfer/Full Access Hypothesis (Schwartz and Sprouse, 1996), which predicts that learners can show native-like processing for novel features. Results also show that learners can establish syntactic dependencies outside of local domains, as suggested by the presence of a P600 for both within and across-phrase violations. Moreover, similar to native speakers, learners were impacted by the structural distance (number of intervening phrases) between the agreeing elements, as suggested by the more positive waveforms for within-phrase than across-phrase agreement overall. These results are consistent with the proposal that learners are sensitive to hierarchical structure.