The letter introduces robust independence tests with non-asymptotically guaranteed significance levels for stochastic linear time-invariant systems, assuming that the observed outputs are synchronous, which means that the systems are driven by jointly i.i.d. noises. Our method provides bounds for the type I error probabilities that are distribution-free, i.e., the innovations can have arbitrary distributions. The algorithm combines confidence region estimates with permutation tests and general dependence measures, such as the Hilbert–Schmidt independence criterion and the distance covariance, to detect any nonlinear dependence between the observed systems. We also prove the consistency of our hypothesis tests under mild assumptions and demonstrate the ideas through the example of autoregressive systems.