Content creation today often takes place via collaborative writing. A longstanding interest of CSCW research lies in understanding and promoting the coordination between co-writers. However, little attention has been paid to individuals who write in their non-native language and to co-writer groups involving them. We present a mixed-method study that fills the above gap. Our participants included 32 co-writer groups, each consisting of one native speaker (NS) of English and one non-native speaker (NNS) with limited proficiency. They performed collaborative writing adopting two different workflows: half of the groups began with NNSs taking the first editing turn and half had NNSs act after NSs. Our data revealed a "late-mover disadvantage" exclusively experienced by NNSs: an NNS's ideational contributions to the joint document were suppressed when their editing turn was placed after an NS's turn, as opposed to ahead of it. Surprisingly, editing help provided by AI-powered tools did not exempt NNSs from being disadvantaged. Instead, it triggered NSs' overestimation of NNSs' English proficiency and agency displayed in the writing, introducing unintended tensions into the collaboration. These findings shed light on the fair assessment and effective promotion of a co-writer's contributions in language diverse settings. In particular, they underscore the necessity of disentangling contributions made to the ideational, expressional, and lexical aspects of the joint writing.