We demonstrate the Monte Carlo calculation of multidimensional tunnel splittings in the water trimer without the use of explicit nodal approximations. The calculations are made for the three-dimensional torsional model of the nonbonded hydrogen atom motions. We employ the projection operator imaginary time spectral evolution (POITSE) method, which requires specification of a set of projector operators. The permutation symmetry leading to exchange tunneling is incorporated into these projectors directly. We analyze the optimal choice of the real space vibrational coordinates with a comparative study of local vs nonlocal symmetry-adapted nuclear coordinates. Extraction of the lowest four sets of tunneling states yields good agreement with available discrete variable representation calculations.