Sexism, deunionization & inequality

Is sexism partly to blame for the decline of trades unions? A new paper suggests so. It finds "a negative and statistically significant link between workplace union density and gender diversity" and says:

The increase in the labour market participation of women in the face of gender discrimination may increase friction between female and male employees within the workplace. This may, in turn, pose challenges in the way of coalition building, which unions need to achieve in order to succeed.

In other words, it might be no accident that the increase in female employment since the 70s has coincided with declining unionization; the former might be one cause (not the only one, of course) of the latter.
Intuitively, we shouldn't be too surprised at this. There's sad evidence that ethnic diversity weakens social solidarity and the building of pro-redistribution coalitions. Mightn't the same be true for gender diversity?
You can read this in two different ways.
You could say that the decline of unions is (partly) due to men's sexism, a reluctance to unionize with women.
Alternatively, you could ascribe it to the increased gender mix of workplaces.
It is, of course, the interaction of the two that does the damage.
This raises an awkward point. The decline of unions since the 70s has contributed (pdf) to rising wage inequality. If increased gender equality (in the sense of more job opportunities for women) has contributed to the decline of unions, then it follows that greater gender equality has been a cause of reduced income equality. Rick recently noted, rightly, that "We are more economically unequal but less socially unequal than we were three decades ago." But he didn't add that the latter might be a cause of the former.In this sense, there are - at least for given (sexist) norms - potential trade-offs between different types of equality.