Synthetic Gaia DR3 surveys from the FIRE cosmological simulations of Milky-Way-mass galaxies

[Abridged] With Gaia Data Release 2, the astronomical community enters a new era of multidimensional surveys of the Milky Way. This new phase-space view of our Galaxy demands new tools for comparing observations to simulations of Milky-Way-mass galaxies in a cosmological context, to test the physics of both dark matter and galaxy formation. We present ananke, a framework for generating synthetic phase-space surveys from high-resolution baryonic simulations, and use it to create a suite of synthetic surveys designed to resemble Gaia DR2 in data structure, magnitude limits, and observational errors. We use three cosmological simulations of Milky-Way-mass galaxies from the Latte suite of the Feedback In Realistic Environments (FIRE) project, which offer many advantages for generating synthetic stellar surveys: self-consistent clustering of star formation in dense molecular clouds, thin stellar and gaseous disks, cosmological accretion and enrichment histories, all in live cosmological halos with satellite dwarf galaxies and stellar halos. We select 3 solar viewpoints from each simulation for a total of 9 synthetic surveys. We generate synthetic stars assuming that each simulation's star particles represent single stellar populations, and distribute synthetic stars in position and velocity using superimposed kernels for different populations. At each viewpoint, we compute a self-consistent dust extinction map from the gas metallicity distribution in each simulation and apply a simple error model to produce a synthetic Gaia-like survey. This results in a catalog of synthetic stars, as if measured by Gaia, that includes both observational properties and a pointer to the generating star particle. We also provide the complete $z = 0$ snapshot--star, gas, and dark matter particles--for each simulated galaxy. We describe data access points, the data model, and plans for future upgrades.