  We introduce and experimentally demonstrate a technique for performing
quantum state tomography 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 sigma-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 quantum state
tomography of systems such as biological molecules where the magnitude and
orientation of the transition dipole moment is not known with high accuracy.
