A search for the central exclusive production of top quark-antiquark pairs (t (t) over bar) is performed for the first time using proton-tagged events in proton-proton collisions at the LHC at a centre-of-mass energy of 13TeV. The data correspond to an integrated luminosity of 29.4 fb(-1). The t (t) over bar decay products are reconstructed using the central CMS detector, while forward protons are measured in the CMS-TOTEM precision proton spectrometer. An observed (expected) upper bound on the production cross section of 0.59 (1.14) pb is set at 95% confidence level, for collisions of protons with fractional momentum losses between 2 and 20%.