  The recently developed framework for quantum theory with no global causal
order allows for quantum processes in which operations in local laboratories
are neither causally ordered nor in a probabilistic mixture of definite causal
orders. The causal relation between the laboratories is described by the
process matrix. We show that, if the inputs of the laboratories are measured in
a fixed basis, one can introduce an effective process matrix which is
operationally indistinguishable from the original one. This effective process
matrix can be obtained by applying the von Neumann- Lu\"ders update rule for
nonselective measurements to the original process matrix and in the bipartite
case it is compatible with a definite causal order. The latter extends the
original Oreshkov et al. proof where one considers that both the measurement of
the input and the re-preparation of the output are performed in a fixed basis.
