Word-piece models (WPMs) are commonly used subword units in state-of-the-art end-to-end automatic speech recognition (ASR) systems. For multilingual ASR, due to the differences in written scripts across languages, multilingual WPMs bring the challenges of having overly large output layers and scaling to more languages. In this work, we propose a universal monolingual output layer (UML) to address such problems. Instead of one output node for only one WPM, UML re-associates each output node with multiple WPMs, one for each language, and results in a smaller monolingual output layer shared across languages. Consequently, the UML enables to switch in the interpretation of each output node depending on the language of the input speech. Experimental results on an 11-language voice search task demonstrated the feasibility of using UML for high-quality and high-efficiency multilingual streaming ASR.