The standard picture of galaxy formation motivates the decomposition of the Milky Way into 3-4 stellar populations with distinct kinematic and elemental abundance distributions: the thin disk, thick disk, bulge, and stellar halo. To test this idea, we construct a Gaussian mixture model (GMM) for both simulated and observed stars in the Solar neighborhood, using measured velocities and iron abundances (i.e., an augmented Toomre diagram) as the distributions to be decomposed. We compare results for the Gaia-APOGEE DR16 crossmatch catalog of the Solar neighborhood with those from a suite of synthetic Gaia-APOGEE crossmatches constructed from FIRE-2 cosmological simulations of Milky Way-mass galaxies. We find that in both the synthetic and real data, the best-fit GMM uses five independent components, some of whose properties resemble the standard populations predicted by galaxy formation theory. Two components can be identified unambiguously as the thin disk and another as the halo. However, instead of a single counterpart to the thick disk, there are three intermediate components with different age and alpha abundance distributions (although these data are not used to construct the model). We use decompositions of the synthetic data to show that the classified components indeed correspond to stars with different origins. By analogy with the simulated data, we show that our mixture model of the real Gaia-APOGEE crossmatch distinguishes the following components: (1) a classic thin disk of young stars on circular orbits (46%), (2) thin disk stars heated by interactions with satellites (22%), (3, 4) two components representing the velocity asymmetry of the alpha-enhanced thick disk (27%), and (5) a stellar halo consistent with early, massive accretion (4%). * Banting Fellow history of the Galaxy we live in (e.g. Freeman & Bland-Hawthorn 2002). In the classic picture, the distribution of stars in velocity and elemental abundances has a relatively small number of distinct components linked to different formation epochs:• At very early times, star formation and proto galaxy merging take place in a relatively chaotic environment, leading to a roughly spheroidal distribution variously