We investigate the relationship among staff engagement, job complementarities and labour supply in the hospital sector, where excessive turnover of the clinical staff (doctors and nurses) can be detrimental for quality of care. We exploit a unique and rich panel dataset constructed by combining employee-level payroll and survey records from the universe of English NHS hospitals. System-GMM estimates remove the endogeneity bias due to reverse causality, revealing nurses' elasticities of retention with respect to engagement of 0.1 and 0.85, and doctors' elasticities of retention with respect to nurses' retention of 0.16 and 0.2, respectively within the hospital and the NHS. Estimates of unconditional quantile regressions confirm these findings, with nurses' engagement-elasticities as large as 1.4 for providers with low retention. Higher engagement is also beneficial to reduce staff absences. Our work is informative on the role played by staff engagement and labour supply complementarities in the workforce planning and management of large organizations.