We introduce and experimentally demonstrate a technique for performing quantum state tomography (QST) on multiple-qubit states despite incomplete knowledge about the unitary operations used to change the measurement basis. Given unitary operations with unknown rotation angles, our method can be used to reconstruct the density matrix of the state up to local σ̌ z rotations as well as recover the magnitude of the unknown rotation angle. We demonstrate high-fidelity self-calibrating tomography on polarization-encoded one- and two-photon states. The unknown unitary operations are realized in two ways: using a birefringent polymer sheet-an inexpensive smartphone screen protector-or alternatively a liquid crystal wave plate with a tuneable retardance. We explore how our technique may be adapted for QST of systems such as biological molecules where the magnitude and orientation of the transition dipole moment is not known with high accuracy.