Due to the dialogue characteristics of unstructured contexts and multi-parties with first-person perspective, many successful text summarization works have failed when dealing with dialogue summarization. In dialogue summarization task, the input dialogue is usually spoken style with ellipsis and co-references but the output summaries are more formal and complete. Therefore, the dialogue summarization model should be able to complete the ellipsis content and co-reference information and then produce a suitable summary accordingly. However, the current state-of-the-art models pay more attention on the topic or structure of summary, rather than the consistency of dialogue summary with its input dialogue context, which may suffer from the personal and logical inconsistency problem. In this paper, we propose a new model, named ReWriteSum, to tackle this problem. Firstly, an utterance rewriter is conducted to complete the ellipsis content of dialogue content and then obtain the rewriting utterances. Then, the co-reference data augmentation mechanism is utilized to replace the referential person name with its specific name to enhance the personal information. Finally, the rewriting utterances and the co-reference replacement data are used in the standard BART model. Experimental results on both SAMSum and DialSum datasets show that our ReWriteSum significantly outperforms baseline models, in terms of both metric-based and human evaluations. Further analysis on multi-speakers also shows that ReWriteSum can obtain relatively higher improvement with more speakers, validating the correctness and property of ReWriteSum.