Compressed sensing enables universal, simple, and reduced-cost acquisition by exploiting a sparse signal model. Most notably, recovery of the signal by computationally efficient algorithms is guaranteed for certain random measurement models, which satisfy the so-called isotropy property. However, in real-world applications, this property is often not satisfied. We propose two related changes in the existing framework for the anisotropic case: (i) a generalized RIP called the restricted biorthogonality property (RBOP); and (ii) correspondingly modified versions of existing greedy pursuit algorithms, which we call oblique pursuits. Oblique pursuits provide recovery guarantees via the RBOP without requiring the isotropy property; hence, these recovery guarantees apply to practical acquisition schemes. Numerical results show that oblique pursuits also perform better than their conventional counterparts.