In my full response to the supposed An Evangelical Statement on the Trinity, which is currently delayed as I have sought feedback to it, I have included a section on Complementarianism. I would like to post that section here first.
We shall start with a brief description of complementarianism, the biblical antidote to egalitarianism. Egalitarianism is the view that equality (especially as it pertains to men and women) pertains to every aspect of being, roles and relationships. Therefore, egalitarianism teaches that there is full equality between men and women to the extent that there are no fundamental differences (besides biological which cannot be denied) between men and women. In society, egalitarianism teaches what is called absolute gender blindness. No discrimination between men and women is allowed, and whatever men can do, women can likewise do. Whatever men can be, women can likewise be. You will seldom hear of course of the converse that men should be able to do what women can do, although egalitarianism should promote that same truth.
In marriage, egalitarianism is worked out in the rejection of any form of headship of the husband over the wife. If any headship is allowed at all (soft egalitarianism), it is a functional headship whereby one party especially the dominant party makes the final decision. In Christian circles, this can be worked out as the husband being the head of the family in the sense of functionally leading the family. Yet any other form of headship is implicitly or explicitly denied.
Complementarianism is the view that agrees that men and women are equal before God. Women are not inferior to men, or vice versa. However, we stress that there is a creational difference of roles and relationships between men and women. Men and women are different not just biologically, but also ontologically as created. If it was just biological, then having a sex-change operation would turn one automatically from a male to a female or vice versa. Rather, gender is not a biological or social construct but a created construct.
Socially, this means that men and women are not the same. Certainly, women should not be discriminated against in terms of denying of jobs and promotions and pay, but this is not what we are getting at. The issue here is that men and women think differently, perceive things differently etc. To the extent that such differences are manifested, they should not be denigrated as mere social constructs, as if women must be able to be almost a carbon copy of men with the exception of biology. Differences between the genders are to be celebrated, not denigrated. Along this line, having mandatory gender quotas for any field of study and work denies both the equality of value and worth, and the created differences between the two sexes. Having a minimum number of women in politics (an affirmative action) is one such error of egalitarianism. Fields such as these should be based purely on interest and meritocracy rather than some misplaced egalitarian view of human nature. If no woman for example wants to join politics, one should not force them to, and one should not discriminate against them entering the field either.
In the family, there is an equality in the worth and value of the marriage partners. But the headship of husband is ordained by God not merely as a functional reality but as a creational ontological reality. A woman submits to her husband not because that is the best way to run a household; not because if both sides insist on their way there would be perpetual conflict. A woman submits to her husband because that is the ordained reality of creation. Similarity in worth and value, but differences in roles and relationships.
In complementarianism, the submission called for is not that of a superior over the inferior. Men are not superior to women and women are not inferior to men. Rather, it is a submission of an equal to another equal, based upon creational reality. Only the embrace of radical egalitarian assumptions would make this incoherent, for egalitarianism equates submission to inferiority, whereas that is not true.