Having multiple children with multiple men.
In our part of the world, Lucy Lawless, Sally Ridge and Wendyl Nissen happily navigate their way through multi-dadding arrangements. ...
Women like Anderson say the negative reaction towards multi-fathered families comes from an assumption that multi-dadding women must be promiscuous.
But Anderson is quick to set the record straight, saying she has only had relationships with four men — the fathers of her four children.
—Shelley Bridgeman, "Who's the daddy?," The New Zealand Herald, September 30, 2007
I have witnessed a broad spectrum experience this week, as I have been on a journey to discover what makes a modern British family.
I have been to visit a group of single mums in Bristol and a "multi-dad" in Derby who claims 18 children by at least five partners.
—Katie Hopkins, "Katie Hopkins' guide to family life," BBC News, August 17, 2007
Sadie Frost has done it. Ulrika Jonsson does it all the time. Paula Yates did it. And so have I. What we, and countless other women have done, is have children with more than one man. It is fraught, it is complicated but, in this day and age, almost inevitable. As more marriages break down and former partners take up with new partners, multi-dadding seems to be a modern phenomenon.
—Lucy Cavendish, "Multi-dads: Having four children by three different men works for me. Is multi-dadding the future of parenting?," The Observer, February 11, 2007
Here's a citation for the adjective multidad, although in this sense it means "having multiple stepfathers":
The play is, however, mildly amusing in a low-impact way. And the writing in the multidad girl's speeches is lovely, full of wonder and humane feeling. ... Sarah Malkin is appealing as the girl for whom Frank is the ninth dad.
—Joe Adcock, "Loony 'Gargoyle' is mildly amusing," The Seattle Post-Intelligencer, August 16, 2003