We report results of a multiband monitoring campaign of the flat spectrum radio quasar TXS 0536+145 at redshift 2.69. This source was detected during a very high gamma-ray activity state in 2012 March by the Large Area Telescope on board Fermi, becoming the gamma-ray flaring blazar at the highest redshift detected so far. At the peak of the flare the source reached an apparent isotropic gamma-ray luminosity of 6.6 x 10^49 erg/s which is comparable to the values achieved by the most luminous blazars. This activity triggered radio-to-X-rays monitoring observations by Swift, Very Long Baseline Array, European VLBI Network, and Medicina single-dish telescope. Significant variability was observed from radio to X-rays supporting the identification of the gamma-ray source with TXS 0536+145. Both the radio and gamma-ray light curves show a similar behaviour, with the gamma-rays leading the radio variability with a time lag of about 4-6 months. The luminosity increase is associated with a flattening of the radio spectrum. No new superluminal component associated with the flare was detected in high resolution parsec-scale radio images. During the flare the gamma-ray spectrum seems to deviate from a power law, showing a curvature that was not present during the average activity state. The gamma-ray properties of TXS 0536+145 are consistent with those shown by the high-redshift gamma-ray blazar population.