The dependent variables in a MANOVA should not be "too strongly correlated". But how strong a correlation is too strong? It would be interesting to get people's opinions on this issue. For instance, would you proceed with MANOVA in the following situations?
Y1 and Y2 are correlated with $r=0.3$ and $p<0.005$
Y1 and Y2 are correlated with $r=0.7$ and $p=0.049$
Some representative quotes in response to @onestop:
"MANOVA works well in situations where there are moderate correlations between DVs" (course notes from San Francisco State Uni)
"The dependent variables are correlated which is appropriate for Manova" (United States EPA Stats Primer)
"The dependent variables should be related conceptually, and they should be correlated with one another at a low to moderate level." (Course notes from Northern Arizona University)
"DVs correlated from about .3 to about .7 are eligible" (Maxwell 2001, Journal of Consumer Psychology)
n.b. I'm not referring to the assumption that the intercorrelation between Y1 and Y2 should be the same across all levels of independent variables, simply to this apparent grey area about the actual magnitude of the intercorrelation.