Ultraperipheral heavy ion collisions constitute an ideal setup to look for exotic hadrons because of their low event multiplicity and the possibility of an efficient background rejection. We propose to look for fourquark states produced by photon-photon fusion in these collisions at the center-of-mass energy per nucleon pair root s(NN) = 5.5 TeV. In particular, we focus on those states that would represent a definite smoking gun for the compact tetraquark model. We show that the X(6900), a likely cc (c) over bar(c) over bar compact state, is a perfect candidate for this search, and estimate a production cross section ranging from around 250 nb to 1150 nb, depending on its quantum numbers. Furthermore, we discuss the importance of ultraperipheral collisions to the search for the scalar and tensor partners of the X(3872) predicted by the diquarkonium model, and not yet observed. The completion of such a flavor-spin multiplet would speak strongly in favor of the compact tetraquark model.