Nonspherical colloids can exhibit liquid-crystalline phases with different degrees of broken orientational and translational symmetry. Here we investigate hard rectangles consisting of photolithographically prepared disks standing on edge. We observe a conventional Kosterlitz-Thouless transition from isotropic to nematic with almost smectic behavior at high density. But just on the isotropic side of the isotropic to nematic transition we observe an unusual regime where short-range tetratic correlations dominate over nematic correlations. This occurs due to the proliferation of Ising-like pi/2 grain boundaries that disrupt nematic order, but preserve tetratic correlations, at lengths shorter than the spacing between free disclinations.