Binning or just skipping values in a simulation to avoid autocorrelation

Given a set of data from a generic Montecarlo simulation $x_i$, $(1=1,...,N)$, autocorrelation is expected to happen between data points within relaxation time $\tau$ (correlation time) distance between each other.

Now, I know that a possibile approach to reduce/avoid correlation is to set up bins much greater than the relaxation time and compute an average from each bin as well as an error.

What if I just consider an initial $x_i$ and then hop to the next one $x_{i+\tau}$ and so on and so forth for $x_{i+2\tau}$ ,$x_{i+3\tau}$ ...essentially considering these new samples as uncorrelated? In this case I'm not computing averages and errors, I'm just skipping enough points to make the remaining ones sort of uncorrelated.

I've read about both approaches in literature, but I'm actually not sure if they are both viable.

Both the approaches you describe are kind-of viable, and should give similar results. Obviously, in the case of just sampling data points at intervals of $\tau$, you are throwing away the intermediate data points; but you may reasonably expect that these do not contain significantly more information, since they are highly correlated with the data points that you do sample, given that $\tau$ is of the order of the correlation time. But I would think that there is nothing to lose by adopting the binning method.