We present a detailed theoretical analysis of the properties and formation of single solitons and higher-order bound dipole pairs in media with anisotropic nonlocal photorefractive material response. The single solitons are elliptical beams, whereas the dipole pairs are formed by a pair of displaced elliptical beams with a rr phase shift between their fields. The theory predicts convergence of Gaussian beams to the solitary states within a certain basin of attraction. Experimental observation of these solitons has been presented elsewhere. The experimental portion of the present paper concentrates on the region further away in parameter space, where complex spatial oscillations, including asymmetric filamentation into several beamlets, occurs. [S1050-2947(97)03011-4].