Bose-Einstein correlations (BEC) in W pair production at e+e- centre-of-mass energies around 200 GeV are studied using data recorded by the OPAL experiment at LEP between 1997 and 2000, yielding a sample of about 7000 WW decays. The data are used to investigate the possible existence of BEC between hadrons originating from different W bosons. The method employed is based on a comparison of the correlation function in fully hadronic WW decays with that constructed from the hadronic part of two semileptonic WW decay events, using an appropriate event-mixing technique. The results are in disagreement with a Monte Carlo model in which it is assumed that identical particles from different W bosons exhibit BEC similar to those observed in single W or Z decays.