How soon after passing the substantial presence test do I owe U.S. tax on my home-country income?

Your worldwide income becomes taxable starting on your residency starting date, which in the year you pass the substantial presence test is generally your first U.S. day that year. Income you earned abroad before that date usually stays outside U.S. tax as part of the nonresident portion of a dual-status year; income after it, including foreign interest, dividends, rent, and salary, is reportable. You may offset foreign taxes with the foreign tax credit to avoid double taxation. This shift is a major reason to nail down your exact residency start. IRS Publication 519 explains residency starting dates and the foreign tax credit at irs.gov. A tax pro is worth it in this transition year.