You might be thinking that polyfidelity is a type of high-end home stereo technology from the late 1970s ... but wait, it's not!
Polyfidelity is a type of polyamory in which the people involved are all in a long-term, stable, committed relationship arrangement. If a polyfidelous relationship goes on long enough, it might look a whole lot like a group marriage, whether or not any of the members of the relationship are legally married in the eyes of whichever state/country they happen to live in. If you've read a lot of Heinlein, this arrangement won't be an alien notion to you.
The term "polyfidelity" was supposedly coined by members of a San Francisco commune, the Kerista Village, which was active from 1971 to 1991. This particular commune expected all its members to be willing to have sex with any/all the other members and to not have relations with anyone outside the commune ... and if that genuinely worked out for them, groovy.
But the actual working polyfidelous households I know of are more flexible and don't look like cults or full-time orgies. They don't look like hippie communes. In fact, they look pretty much like any other ordinary household of roommates banding together to save on rent; for every vocal, flaming polyamorist in the world, there are a hundred more quietly going about their lives. In many communities it is deeply unwise to advertise one's poly status, especially if a person has, say, a job as a professor at a Catholic university, or has custody of children from a previous marriage.
The main thing about polyfidelity is that the people involved care about and respect each other, consider each other equals, and have made a long-term commitment to making things work as a household or set of linked households. It's not all about the sex; some polyfidelous relationships may be involved in co-parenting, or running a business, or any of a number of other endeavors that don't involve nookie. If someone wants to date someone outside the core group, often the other members will have veto power ("You want to go out with Rick? That guy would go down on the Titanic and also he stole my limited-edition Catan game three years ago; I don't trust him.") with the understanding that they'll be reasonable about it and not simply turn down all prospective secondaries out of petty jealousy.
I know of a poly household here in Columbus; the people have been together for years. Alice and Bob are legally married to each other; they live in a house with Christine, who they are both romantically involved with. In the house next door live Doug and Eve, who are legally married to each other; D&E are also Canadian citizens and have to head back up north of the border every so often for residency purposes. So, they have a house in Canada that is occupied full-time by Freddie, who is a sweetie of Eve's and a good friend of Doug's. Alice and Bob own the neighboring house Doug and Eve live in; Alice and Eve are involved romantically, but Eve is not sleeping with Doug or Christine, and Doug and Bob are platonic friends who hang out all the time playing video games.
Yeah. At some point you really do need a flowchart to keep track of everyone, but it works for them, and has for quite some time. Occasionaly one of them will start dating someone outside their household, and the others can say "I'd really rather you didn't, and here's a rational reason why" and the veto will be abided by. Or there will be more discussion and negotiation and the veto might be retracted. Mostly, though, the people in the household seem content with their existing relationships, because dating takes a lot of time and energy and they all have jobs etc. to worry about. Furthermore, any new romantic entanglement generates the risk of creating new, damaging drama unless they're very careful about their choice of new partners.
I know of other examples of relationships that I think "count" as polyfidelity, even though there is not an intensive sharing of households. Gail and Hector are legally married and were polyamorous before their marriage; shortly after they married, Gail started seeing Inigo, who is a busy researcher at a nearby college. Fifteen years into the marriage, Hector and Gail now have three children, one of whom was fathered by Inigo with Hector's blessing. Inigo is over at their house often and is accepted as family by all the children. The three adults check in with each other before they start dating anyone else and regularly get tested for STDs, and the needs of the children always come first.
The benefit of closed polyfidelity -- the members see each other, but don't date outside their group -- is that once everybody has been tested for STDs and gets clean bills of health, they pretty much don't have to worry about getting tested again. Physical health aside, people involved in polyfidelity can often have deeper, more emotionally-rewarding relationships and a better sense of security than people involved in casual polyamory or casual serial monogamy. And on a practical level, a household of five working adults is a lot better able to weather layoffs and other financial misfortunes than a household of two adults.
The downside? Maintaining functional polyfidelity is quite a lot of work, and is not for people who are unwilling to compromise and communicate with integrity. Marriage of any kind is challenging; if a person can't responsibly manage a relationship with just one other person, it's likely to be utterly disastrous with two or three. People in polyfidelous households have to deal with the same marital conflicts everybody else does: disputes over neglected chores, money, kids ... all of that mundane stuff can still blow up into a huge problem. And while having more people around can sometimes help diffuse conflict, sometimes the conflict is amplified. And when a breakup happens between a couple of members, it can destroy the entire group. You think divorce between two people can get bad? Throw in two or three more people and their kids along with ambiguous legal statuses and it's so much worse.