First, some definitions:
Coded Misogyny refers to the hatred or marginalization of something on the basis that it is construed to be feminine, though not explicitly so. If we think of "Women are too emotional" as overt misogyny used to demean both individual and all women, coded misogyny builds on that sort of basic bias to demean and undervalue people, pursuits, fashions, habits, vocations, institutions, etc. that are coded as feminine, e.g. teaching, nursing, paid childcare and eldercare are fields disproportionately staffed by women and are thus endemically underpaid or undervalued careers.
Institutional Prejudice refers to the debasement of an entire career, organization, pastime, or some other collective on the basis that it is coded as feminine.
So: What this is about is the insidiousness of coded misogyny and how some of the most familiar institutional prejudice is attributable to coded misogyny.
Sometimes, coded misogyny is not merely about someone/something being "too feminine," but about someone/something being "not feminine enough." Much of the institutional prejudice regarding gender and sexuality that deviates from the Patriarchy-approved binary is coded misogyny: Gay and trans* men are too feminine; lesbian and trans* women are not feminine enough.
And frequently, coded misogyny intersects with other coded oppressions: Women's corporate attire, for example, is a kyriarchal carousel of intersectional prejudice. A women's business suit is not defined by the fact that a woman is wearing it, but by its tailoring, designed to emphasize a particular ideal female form that is meant to be underneath. A woman wearing a suit tailored for a man would be considered transgressive and thus unacceptable in most corporate environments, even if her body were more flattered by a men's suit.
Black women, and other women of color with kinky hair, also face strong disincentives against wearing their hair naturally, which is coded to be indicative of radical womanhood. A white woman who wears her hair long and grey and unstyled is more likely to be coded as a women's studies professor than a corporate executive.
All of which happens inside an environment which is coded masculine to begin with—which is why corporate work is considered serious and important, while the arts, which are coded feminine, are considered unserious and superfluous.
Which, in turn, is why the National Endowment for the Arts (feminine) is constantly in threat of being defunded, but solemn discussion about reducing the budget for the Defense Department (masculine) is considered a hilarious suggestion, particularly by the Republicans, who are the "Daddy Party," strong and inflexible and warlike, as opposed to the "Mommy Party" Democrats, who are ostensibly soft (on defense) and empathetic and diplomatic.
"Softness," of any description, is coded feminine, and thus is anything deemed soft likely to be the victim of institutional prejudice. This can be something metaphorically soft, like a foreign policy that favors negotiation over aggression, or something that is literally soft, like a fat body.
There are a lot of reasons for fat prejudice, but one of them, especially in a misogynistic culture, is that fat tends to exaggerate female characteristics in (especially cis) female bodies. Bigger boobs, bigger hips, bigger ass. Some fat women look like they might be pregnant. Fat can exaggerate the female form and female sexuality. It also tends give female-coded characteristics to (especially cis) male bodies, evidenced even in the way fat men are mocked—they have "man boobs" and their beer bellies are construed as pregnancies for jokes like, "When are you due, Frank?"
And then there is the stereotype of fat people as lazy, as couch potatoes; it is not incidental that "active" is coded masculine and "passive" is coded feminine in a culture that disdains fat people and axiomatically views them as not active. Fat people are softer, their bodies considered more feminine, their habits coded feminine. Fat hatred is thus, like many other oppressions, an institutional prejudice with roots in coded misogyny.
The active/passive binary is hardly the only one we code as masculine/feminine. Rational/empathetic (or irrational) are also coded masculine/feminine, which is one of the most pernicious binaries in institutional prejudice. See aforementioned divide between how conservatives (rational) and progressives (empathetic) are viewed.
Science is coded rational/masculine; spirituality is coded irrational/feminine. I'm an atheist, but one of my many problems with the nascent "atheist movement" is the enormous amount of coded misogyny one finds in the dismissal of spiritualism. It isn't impossible to discuss one's problems with religion without coded misogyny, but it is contingent on rejecting the language of a deeply entrenched binary embedded with coded misogyny.
So, too, discussions of Western/non-Western medicine. Western medicine is coded rational/masculine; alternative medicine is coded irrational/feminine. And here again is an example of intersectional prejudice, as many alternative practices (yoga, acupuncture, acupressure, massage, herbal remedies) are imported from the East. (And practiced disproportionately by people marginalized in the West: Women/people of color/queers.) The wholesale dismissal of alternative practitioners and practices as quacks and opportunists (despite there being plenty of quacks and opportunists to be found in Western medicine, too) is an institutional prejudice frequently loaded with both coded misogyny and coded racism. Which is a particular misfortune for people with marginalized bodies, because, as I've had occasion to mention before, there's A LOT that Western medicine gets wrong, especially when it comes to treating people who aren't straight white thin generally able-bodied cis men.
Also no coincidence: That much of the alleged "witchcraft" that was used to justify literal witch-hunts, in which extraordinary numbers of women have died, was alternative medicine and/or practices with exclusive relevance for (typically cis) female bodies. Midwifery was considered witchcraft once upon a time for this very reason.
The examples are endless: Monotheistic religions with a male god (and institutional misogyny, ahem) are similarly privileged over female-centric religions. Believers in bootstraps are privileged over believers in a social safety net, because authoritarianism and individualism are coded as masculine while socialism and collectivism are coded as feminine. Meat-eating (manly!) is privileged over vegetarianism and veganism (salads are for girls!). Et cetera, et cetera, et cetera.
What you'll note about how people in the marginalized groups are dismissed, irrespective of their individual gender—wacky, woo-woo, unserious, irrational, emotional, empathetic, weak—are the same way women themselves are often dismissed.
That, also, is no coincidence.
When we hear people dismissing whole slices of culture, without caveat or exception, using the same sorts of language that misogynists use to dismiss women—or when we catch ourselves using that language, thus more deeply entrenching the ubiquitous trope that anything coded feminine is inherently less than—we should question why that is.
Just as we question why specific classes of people are privileged over another, we must question why specific institutions are privileged over others, and how that privileging might actually exist in service to a patriarchy.
It's not difficult, upon close examination, to understand how uncritically privileging Catholicism, which is male-centric and authoritarian, over a female-centered paganism serves the narratives of the Patriarchy. It is not difficult, upon close examination, to understand how reflexively privileging Western medicine, which is best at serving the needs of financially privileged straight white thin generally able-bodied cis men, serves the narratives of the Patriarchy. The point isn't that this stuff isn't evident upon reflection, but that we aren't predisposed to think about how coded misogyny works on an institutional scale.
Nor how that then serves to reinforce the oppression and marginalization of individual women.
Because as long as the narrative that "anything coded feminine is inherently less than" is allowed to flourish on any scale, women cannot be truly equal.
This is not, of course, an argument for regarding a demonstrable snake-oil salesman the same way we regard an ethical and principled scientist. It is, as are most feminist arguments, an argument for nuanced thinking, for not looking at the world in black and white binaries, for not letting assumptions about institutions stand in substitute for facts about the individuals who populate them, and, most importantly, for avoiding gender essentialist binaries when discussing those institutions, a habit which starts by being aware of those binaries and how they work in the first place.