This paper describes a semi-supervised system that jointly learns verbal multiword expressions (VMWEs) and dependency parse trees as an auxiliary task. The model benefits from pre-trained multilingual BERT. BERT hidden layers are shared among the two tasks and we introduce an additional linear layer to retrieve VMWE tags. The dependency parse tree prediction is modelled by a linear layer and a bilinear one plus a tree CRF architecture on top of the shared BERT. The system has participated in the open track of the PARSEME shared task 2020 and ranked first in terms of F1-score in identifying unseen VMWEs as well as VMWEs in general, averaged across all 14 languages.