  We introduce a moment map picture for holomorphic string algebroids where the
Hamiltonian gauge action is described by means of inner automorphisms of
Courant algebroids. The zero locus of our moment map is given by the solutions
of the Calabi system, a coupled system of equations which provides a unifying
framework for the classical Calabi problem and the Hull-Strominger system. Our
main results are concerned with the geometry of the moduli space of solutions,
and assume a technical condition which is fulfilled in examples. We prove that
the moduli space carries a pseudo-K\"ahler metric with K\"ahler potential given
by the dilaton functional, a topological formula for the metric, and an
infinitesimal Donaldson-Uhlenbeck-Yau type theorem.
