  We study quantum tomography based on a stochastic continuous-time measurement
record obtained from a probe field collectively interacting with an ensemble of
identically prepared systems. In comparison to previous studies, we consider
here the case in which the measurement-induced backaction has a nonnegligible
effect on the dynamical evolution of the ensemble. We formulate a maximum
likelihood estimate for the initial quantum state given only a single instance
of the continuous diffusive measurement record. We apply our estimator to the
simplest problem -- state tomography of a single pure qubit, which, during the
course of the measurement, is also subjected to dynamical control. We identify
a regime where the many-body system is well approximated at all times by a
separable pure spin coherent state, whose Bloch vector undergoes a conditional
stochastic evolution. We simulate the results of our estimator and show that we
can achieve close to the upper bound of fidelity set by the optimal POVM. This
estimate is compared to, and significantly outperforms, an equivalent estimator
that ignores measurement backaction.
