Structural nested mean models (SNMMs) are among the fundamental tools for inferring causal effects of time-dependent exposures from longitudinal studies. With binary outcomes, however, current methods for estimating multiplicative and additive SNMM parameters suffer from variation dependence between the causal SNMM parameters and the non-causal nuisance parameters. Estimating methods for logistic SNMMs do not suffer from this dependence. Unfortunately, in contrast with the multiplicative and additive models, unbiased estimation of the causal parameters of a logistic SNMM rely on additional modeling assumptions even when the treatment probabilities are known. These difficulties have hindered the uptake of SNMMs in epidemiological practice, where binary outcomes are common. We solve the variation dependence problem for the binary multiplicative SNMM by a reparametrization of the non-causal nuisance parameters. Our novel nuisance parameters are variation independent of the causal parameters, and hence allows the fitting of a multiplicative SNMM by unconstrained maximum likelihood. It also allows one to construct true (i.e. congenial) doubly robust estimators of the causal parameters. Along the way, we prove that an additive SNMM with binary outcomes does not admit a variation independent parametrization, thus explaining why we restrict ourselves to the multiplicative SNMM.