An estimated 5·1 million stillbirths and neonatal deaths occur annually. Household surveys, most notably the Demographic and Health Survey (DHS), run in more than 90 countries and are the main data source from the highest burden regions, but data-quality concerns remain. We aimed to compare two questionnaires: a full birth history module with additional questions on pregnancy losses (FBH+; the current DHS standard) and a full pregnancy history module (FPH), which collects information on all livebirths, stillbirths, miscarriages, and neonatal deaths. Women residing in five Health and Demographic Surveillance System sites within the INDEPTH Network (Bandim in Guinea-Bissau, Dabat in Ethiopia, IgangaMayuge in Uganda, Matlab in Bangladesh, and Kintampo in Ghana) were randomly assigned (individually) to be interviewed using either FBH+ or FPH between July 28, 2017, and Aug 13, 2018. The primary outcomes were stillbirths and neonatal deaths in the 5 years before the survey interview (measured by stillbirth rate [SBR] and neonatal mortality rate [NMR]) and mean time taken to complete the maternity history section of the questionnaire. We also assessed between-site heterogeneity. This study is registered with the Research Registry, 4720. 69 176 women were allocated to be interviewed by either FBH+ (n=34 805) or FPH (n=34 371). The mean time taken to complete FPH (10·5 min) was longer than for FBH+ (9·1 min; p<0·0001). Using FPH, the estimated SBR was 17·4 per 1000 total births, 21% (95% CI -10 to 62) higher than with FBH+ (15·2 per 1000 total births; p=0·20) in the 5 years preceding the survey interview. There was strong evidence of between-site heterogeneity (I2=80·9%; p<0·0001), with SBR higher for FPH than for FBH+ in four of five sites. The estimated NMR did not differ between modules (FPH 25·1 per 1000 livebirths vs FBH+ 25·4 per 1000 livebirths), with no evidence of between-site heterogeneity (I2=0·7%; p=0·40). FPH takes an average of 1·4 min longer to complete than does FBH+, but has the potential to increase reporting of stillbirths in high burden contexts. The between-site heterogeneity we found might reflect variations in interviewer training and survey implementation, emphasising the importance of interviewer skills, training, and consistent implementation in data quality. Children's Investment Fund Foundation.