  We consider the problem of non-parametric testing of independence of two
components of a stationary bivariate spatial process. In particular, we revisit
the random shift approach that has become a standard method for testing the
independent superposition hypothesis in spatial statistics, and it is widely
used in a plethora of practical applications. However, this method has a
problem of liberality caused by breaking the marginal spatial correlation
structure due to the toroidal correction. This indeed causes that the
assumption of exchangability, which is essential for the Monte Carlo test to be
exact, is not fulfilled.
  We present a number of permutation strategies and show that the random shift
with the variance correction brings a suitable improvement compared to the
torus correction in the random field case. It reduces the liberality and
achieves the largest power from all investigated variants. To obtain the
variance for the variance correction method, several approaches were studied.
The best results were achieved, for the sample covariance as the test
statistics, with the correction factor $1/n$. This corresponds to the
asymptotic order of the variance of the test statistics.
  In the point process case, the problem of deviations from exchangeability is
far more complex and we propose an alternative strategy based on the mean cross
nearest-neighbor distance and torus correction. It reduces the liberality but
achieves slightly lower power than the usual cross $K$-function. Therefore we
recommend it, when the point patterns are clustered, where the cross
$K$-function achieves liberality.
