We use numerical bootstrap techniques to study correlation functions of traceless sym-metric tensors of O(N) with two indices ti j. We obtain upper bounds on operator dimen-sions for all the relevant representations and several values of N. We discover several families of kinks, which do not correspond to any known model and we discuss possi-ble candidates. We then specialize to the case N = 4, which has been conjectured to describe a phase transition in the antiferromagnetic real projective model ARP3. Lattice simulations provide strong evidence for the existence of a second order phase transition, while an effective field theory approach does not predict any fixed point. We identify a set of assumptions that constrain operator dimensions to a closed region overlapping with the lattice prediction. The region is still present after pushing the numerics in the single correlator case or when considering a mixed system involving t and the lowest dimension scalar singlet.