It has recently been shown that a subdominant hidden sector of atomic dark matter in the early Universe provides a novel avenue toward resolving the Hubble ($H_0$) tension while maintaining good agreement with cosmic microwave background era observables. However, such a mechanism requires a hidden sector whose energy density ratios are the same as in our sector and whose recombination also takes place at redshift $z≈1100$, which presents an apparent fine-tuning. We introduce a realistic model of this scenario that dynamically enforces these coincidences without fine tuning. In our setup, the hidden sector contains an identical copy of Standard Model (SM) fields but has a smaller Higgs vacuum expectation value (VEV) and a lower temperature. The baryon asymmetries and reheating temperatures in both sectors arise from the decays of an Affleck-Dine scalar field, whose branching ratios automatically ensure that the reheating temperature in each sector is proportional to the corresponding Higgs VEV. The same setup also naturally ensures that the hydrogen binding energy in each sector is proportional to the corresponding VEV, so the ratios of binding energy to temperature are approximately equal in the two sectors. Furthermore, our scenario predicts a correlation between the SM/hidden temperature ratio and the atomic dark matter abundance and automatically yields values for these quantities favored by concordant early- and late-Universe measurements of $H_0$.