How do egalitarians understand male headship?

Summary

Egalitarians emphasize the "one another" meaning of Eph 5:21 ("... submitting to one another out of reverence for Christ.") to control the interpretation of Eph 5:21-33 while being silent on Col 3:18 and 1 Pet 3:1. Secondly, they interpret "head" (kephal) to mean "source" instead of "authority over".

The Longer Answer

It's interesting how several English translations (such as NLT, NIV, NRSV) put Eph 5:21 in the same pericope with Eph 5:21-33 while other English translations (such as ESV,NASB, ASV, NKJV) put Eph 5:21 in the earlier pericope Eph 5:1-21. Grouping Eph 5:21 with Eph 5:21-33 enables one to apply "submit to one another" to the husband and wife relationship.

An example of a proponent of egalitarian interpretation of Eph 5:21 is in a 2015 blog article Husbands, Crucify Your Privilege: Ephesians 5 & How Men Take the Lead by Kurt Willems, the founding lead pastor of Pangea Church in Seattle, who distinguishes Patriarchy (exemplified by the Roman culture at the time of St. Paul), Complementarianism, and Egalitarianism/Mutuality. The argument emphasizes the "each other" in 5:21 as two directional submission and paraphrases Eph 5 as:

“Husbands, in this Roman culture men have all the privilege, but in submission out of reverence for Christ, you are invited to take the lead in crucifying your privilege. In doing so, you look like Jesus and can be part of a marriage of mutual image-bearing and kingdom co-leadership.”

Another example is the Junia Project advocating the full inclusion of women in church leadership and for mutuality in marriage. The website links to a paper The Bible Teaches the Equal Standing of Man and Woman by Philip B. Payne holding a PhD in New Testament Studies from Cambridge and who wrote a 2009 book published by Zondervan Man and Woman, One in Christ reviewed here which summarizes his egalitarian exegesis of relevant Bible passages. From the review:

Therefore, Paul appropriated, according to Payne, the egalitarian vision enunciated in Gen. 1–2 and the rest of the Old Testament Scriptures. Galatians 3:28 proclaims the equality of men and women in Christ, and the text cannot be restricted to soteriology. Social and cultural implications follow from Paul’s bold declaration in Galatians. The fundamental equality of married couples is confirmed by 1 Cor. 7. Ephesians 5:22–33 in no way contradicts this since the text teaches mutual submission (Eph. 5:21). Furthermore, the word kephal? is understood to mean “source” instead of “authority over.”

Mr. Payne's paper's opening paragraph:

Is the Bible divided on the issue of gender? Many highly respected evangelical scholars believe there is a tension in the Bible between affirmations of gender equality and gender roles. Can we arrive at a consistent biblical position without doing violence to the text? Need one sacrifice good exegesis at the altar of systematic theology? Surely, good exegesis and good systematic theology go hand in hand. I have prayerfully wrestled for forty-one years with the texts’ apparent contradictions on gender and can honestly say that the biblical texts themselves have transformed my understanding. From creation to new creation, the Bible’s message about gender in the church and marriage consistently affirms the equal standing of man and woman.

With regards to Ephesians as well as the church in Ephesus, Mr. Payne made the following points in the paper:

• Paul's affirmations of the equality of husband and wife: after discussing 1 Cor 7 as "Paul's most detailed treatment of marriage" where Paul "repeatedly uses symmetrically-balanced wording to reinforce this equality" he continues:

  Similarly, in Eph 5:21–22, the wife’s submission is explicitly one facet of mutual submission, each voluntarily yielding in love.23 Paul’s call to both wives and husbands is to defer to and nurture one another. Christ is the model for all believers, wives as well as husbands (5:2). Paul defines what he means by Christ being “head” in verse 23 by equating it with “savior” through emphatic apposition: “Christ the head of the church, he the savior of the body.” What does Christ do as “savior”? Paul explains: “Christ gives himself” for the church (5:25) and “nourishes and cherishes” it (5:29). This shows that Paul is using “head” with the established Greek meaning “source,” here focusing on Christ as the source of love and nourishment of the church.24 Paul calls husbands as “head” of the wife to follow Christ’s example as “head” by loving, nourishing, and cherishing their wives (5:25–29). This cherishing, nourishing love, not a hierarchy of authority, motivates her submission (5:23).

  The Bible approves women leading in the home. Paul treats husbands and wives equally in relation to their children (Eph 6:1–2; Col 3:20) and tells wives to “rule their households” (literally, “be house despots,” 1 Tim 5:14). If this is not leadership in the home, what is?

• 1 Timothy 2:12: he argues that here Paul is not prohibiting women from teaching or having authority over men but "because of the ongoing crisis of false teaching in Ephesus, he prohibits women from unauthorized assumption of authority over a man." Furthermore, Paul is not prohibiting Priscilla who was present in Ephesus (see 2 Tim 4:19) from assuming properly delegated or recognized authority or how she "explained to [Apollos] the way of God more accurately (Acts 18:26).

Further study

A 2009 issue of Christian Reflection (a quarterly publication of Baylor University's Center for Christian Ethics): Women and the Church (pdf here) has a book review essay Women in Ministry: Beyond the Impasse by Gretchen E. Ziegenhals covering 3 recent books, one of them is Zondervan's Two Views on Women in Ministry which includes 4 evangelical NT scholars holding seminary faculty positions with 2 arguing for the egalitarian position (Craig S. Keener and Linda L. Belleville) and 2 arguing for the complementarian position (Thomas R. Schreiner and Craig L. Blomberg).