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The anticorrelated quantum fluctuations of the two lasing modes of a VCSEL is studied. The behaviour of the fluctuations of TEM00 and TEM01 transverse modes and their correlation is investigated by measuring the transverse spatial distribution of intensity noise of the laser beam. Results are compared with predictions of a full quantum model which include the transverse structure of the lasing modes as well as the spatial correlations created by the competition of different transverse modes. The model includes a spatial dependence of the pump and allows for the simultaneous lasing of two transverse modes. The quantum fluctuations are calculated from the Heisenberg Langevin equations of the system.