We investigate the clustering and halo occupation distribution (HOD) of DESI Y1 emission-line (ELGs) and luminous red (LRGs) galaxies at 0.8 < z < 1.1, including their cross-correlation (ELG×LRG), using the ABACUSSUMMIT suite and a new Halo Occupation Model (HOMe) for galaxy multitracers. This integrates intrahalo dynamics, halo exclusion, and quenching, bridging insights from hydrodynamical, HOD, abundance-matching, and semianalytic studies. Leveraging full phase-space information from the UCHUU N-body simulation, and sampling satellites from dark-matter particle positions via physically motivated prescriptions, HOMe reproduces the anisotropic clustering down to s = 200 h−1 kpc with unprecedented accuracy. Model parameters are inferred solely from two-point statistics using a two-level Bayesian framework, yielding high-fidelity ELG, LRG, and cross-reference catalogs. We find that satellite ELGs behave as incoherent flows within their parent halos, dominating the clustering below 4 h−1 Mpc. The HOD from the best-fit HOMe has the following properties: (i) 90.50% (85.91%) of ELGs (LRGs) are central galaxies without satellites, residing in halos of Mvir ∼ 6.6 × 1011 (1.2 × 1013) h−1M⊙; (ii) the ELG×LRG cross-correlation is governed by central-central pairs and shaped by halo exclusion on 2─5 h−1 Mpc scales; (iii) 9.50% (14.09%) of ELGs (LRGs) are satellites, of which 1.09% (3.52%) inhabit halos with a central galaxy of the same species in a maximally conformal configuration, 7.02% (0.005%) orbit complementary hosts in a minimally conformal state, and 0.58% (10.57%) are orphans. The high sensitivity of HOMe precisely captures the dynamics of satellites in different host environments, opening a promising avenue for understanding systematics and the dynamical nature of dark matter, potentially distinguishing gravity models.