We develop in detail a recently proposed optical-path modification of astronomical intensity interferometers. Extended-path intensity correlation (EPIC, see companion paper) introduces a tunable path extension, enabling differential astrometry of multiple compact sources such as stars and quasars at separations of up to a few arcseconds. Combined with other recent technological advances in spectroscopy and fast single-photon detection, a ground-based intensity interferometer array can achieve microarcsecond resolution and even better light-centroiding accuracy on bright sources of magnitude $m\lesssim 15$. We lay out the theory and technical requirements of EPIC, and discuss the scientific potential. Promising applications include astrometric lensing of stars and quasar images, binary-orbit characterization, exoplanet detection, Galactic acceleration measurements and calibration of the cosmic distance ladder. The introduction of the path extension thus significantly increases the scope of intensity interferometry while reaching unprecedented levels of relative astrometric precision.

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• Optical interferometry
• Optical, UV, & IR astronomy
• Binary stars
• Extrasolar planets
• Galactic nuclei & quasars
• Stars
• Astronomical masses & mass distributions
• Telescopes