I want to fit a mixture model with continuous (input) variables to cluster my data. Some of the variables are correlated with each other. Should I remove the correlated variables and retain only one of them (per group of correlated variables) or can I keep all of them?
If they are very strongly correlated, the result will likely be better when you remove them first (just as with any known trend in the data that you do not want to discover, actually).
Other than that, the classic EM Gaussian Mixtures clustering can deal with linear correlations quite well, as the covariance matrix will adjust the distribution to that.