Until quite recently, extending Phrase-based Statistical Machine Translation (PBSMT) with syntactic structure caused system performance to deteriorate. In this work we show that incorporating lexical syntactic descriptions in the form of supertags can yield significantly better PBSMT systems. We describe a novel PBSMT model that integrates supertags into the target language model and the target side of the translation model. Two kinds of supertags are employed: those from Lexicalized Tree-Adjoining Grammar and Combinatory Categorial Grammar. Despite the differences between these two approaches, the supertaggers give similar improvements. In addition to supertagging, we also explore the utility of a surface global grammaticality measure based on combinatory operators. We perform various experiments on the Arabic to English NIST 2005 test set addressing issues such as sparseness, scalability and the utility of system subcomponents. Our best result (0.4688 BLEU) improves by 6.1% relative to a state-of-theart PBSMT model, which compares very favourably with the leading systems on the NIST 2005 task.