We report on the first joint search for gravitational waves by the TAMA and LIGO collaborations. We looked for millisecond-duration unmodelled gravitational-wave bursts in 473 hr of coincident data collected during early 2003. No candidate signals were found. We set an upper limit of 0.12 events per day on the rate of detectable gravitational-wave bursts, at 90% confidence level. From simulations, we estimate that our detector network was sensitive to bursts with root-sum-square strain amplitude above approximately 1-3x10^{-19} Hz^{-1/2} in the frequency band 700-2000 Hz. We describe the details of this collaborative search, with particular emphasis on its advantages and disadvantages compared to searches by LIGO and TAMA separately using the same data. Benefits include a lower background and longer observation time, at some cost in sensitivity and bandwidth. We also demonstrate techniques for performing coincidence searches with a heterogeneous network of detectors with different noise spectra and orientations. These techniques include using coordinated signal injections to estimate the network sensitivity, and tuning the analysis to maximize the sensitivity and the livetime, subject to constraints on the background.
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