  The non-relativistic effective theory of dark matter-nucleon interactions
depends on 28 coupling strengths for dark matter spin up to 1/2. Due to the
vast parameter space of the effective theory, most experiments searching for
dark matter interpret the results assuming that only one of the coupling
strengths is non-zero. On the other hand, dark matter models generically lead
in the non-relativistic limit to several interactions which interfere with one
another, therefore the published limits cannot be straightforwardly applied to
model predictions. We present a method to determine a rigorous upper limit on
the dark matter-nucleon interaction strength including all possible
interferences among operators. We illustrate the method to derive model
independent upper limits on the interaction strengths from the null search
results from XENON1T, PICO-60 and IceCube. For some interactions, the limits on
the coupling strengths are relaxed by more than one order of magnitude. We also
present a method that allows to combine the results from different experiments,
thus exploiting the synergy between different targets in exploring the
parameter space of dark matter-nucleon interactions.
