  We develop in detail a recently proposed optical-path modification of
astronomical intensity interferometers. Extended-Path Intensity Correlation
(EPIC) 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.
