How do the correlations between Big 5 personality change in studies examining "faking bad"? I have conducted several studies that get participants to complete the IPIP personality test while role play going for a job. A common finding is that the average correlation (i.e., $\bar{r}$ between big 5 personality scales increases the more you ask people to fake. 
For example, the rough pattern in the data is:


Get participants to complete a personality test honesty ($\bar{r} = .25$)
Get participants to complete do a job applicant role-play ($\bar{r} = .40$)
Get participants to complete a job applicant role-play and explicitly tell then to fake it ($\bar{r} = .60$)


Thus, increasing the amount of faking, increases the average intercorrelation.
I'm interested in what happens when participants "fake bad" (e.g., specifically try to present a negative impression such as depressed, disturbed, or socially undesirable).


What have previous studies found to be the effect of "faking bad" on the average scale intercorrelations of Big 5 personality?
Have any researchers offered theoretical explanations for any observed effect of faking bad on such changes average scale intercorrelations?
I don't know any actual "faking bad" studies or theories involving the Big Five, but there are some theories about the Big Five's general factor that can both explain the common finding you mention and inform a hypothesis about potential results from a study that would address your questions. I don't expect this will be a very satisfying answer, because I'm offering it in a bit of a hurry and referencing your own questions (and answer) to provide referential substance, but here it is anyway.

As your question and answer point out, there is a general factor underlying the Big Five, an even bigger "Big One." It partly reflects social desirability, as you know from your other question, and partly reflects people's efforts to align their personalities (or at least their opinions about their own personalities) toward those more desirable poles of the dimensions. It may even reflect some adaptive advantage to being slightly toward the desirable side of average (though probably not all the way to the extreme end of that side). Whether people succeed at being these desirable, maybe better-adjusted versions of themselves in the long term is immaterial. They try in the short term, and tell themselves it's working for various reasons, including self-serving bias and self-perceptive processes in identity (that inform judgments with recent behaviors).

People have some intuitive sense of the desirability of being sociable, stable, friendly, responsible, and curious (not to mention other aspects of each trait). If asked to act like a professionally desirable (i.e., hire-worthy) applicant, they will bias their answers in these socially desirable directions. These biases (whether intentionally deceptive or unconsciously ego-defensive) partly overlay some of the natural, independent variation with an internally consistent personality profile that suits the social consensus of what is desirable. Therefore people seem more uniform and correlations strengthen. Maybe you know other explanations of this finding as well; this may be one explanation, but I doubt it's the only.

I think this explanation could apply equally well if people were asked to "fake bad." Again, a socially undesirable personality profile is socially salient and consensual enough to overlay natural, independent variation with a relatively uniform pattern of bias. Most people know you're probably less likely to get hired if you act shy, edgy, hostile, impulsive, and stubborn (again, among other things that generally point in the same overall directions more often than not). Therefore I'd expect increased correlations in the "faking bad" scenario too.

Not a well-researched theory here I'm afraid; just a hypothesis or half-educated guess. I hope it's helpful nonetheless, and would be happy to look further into it and edit if anyone's interested enough to comment on a pseudo-answer that's nine months late. I can definitely offer some references for at least some of the instrumental points of theory I offer here without explicit support if that's socially desirable enough!