We present a transition-based system that jointly predicts the syntactic structure and lexical units of a sentence by building two structures over the input words: a syntactic dependency tree and a forest of lexical units including multiword expressions (MWEs). This combined representation allows us to capture both the syntactic and semantic structure of MWEs, which in turn enables deeper downstream semantic analysis, especially for semi-compositional MWEs. The proposed system extends the arc-standard transition system for dependency parsing with transitions for building complex lexical units. Experiments on two different data sets show that the approach significantly improves MWE identification accuracy (and sometimes syntactic accuracy) compared to existing joint approaches.