This is a personal observation post and not my traditional aggregation of studies. Your experience may differ.
- In US (does not ship outside at last query): https://pendulumlife.com/products/pendulum-akkermansia-3?view=akkermansia
- Outside of US: https://www.desertcart.ae/products/401266402-akkermansia-muciniphila-25-billion-cfu-p
This is available exclusively as Pendulum Akkermansia. We signed up for the subscription plan when it was first announced, and they have kept shipping at their low introductory price every month. Within I week of getting the first shipment, I read a few posts that “it did not persist”. Excuse me, the typical turnaround time for a microbiome test is 3-4 weeks. I suspect that someone is gas-lighting. Or, they may have been taking a hostile probiotic at the same time.
We subscribed to only one bottle and we decided that the wife would take it for the first two months (we rotate probiotics). The results were her Akkermansia muciniphila went from 4,530 to 64,920 (91%ile) — so, it appears to have persisted (or helped the native ones to prosper by creating a friendlier environment). I should point out that that the fact that she has some to start with may be a significant factor for this awesome increase (that is, the environment was not totally hostile).
She noticed some significant changes, to her the most significant one was a major improvement in sleep.
Kampo (Japanese Name)
This is also known as fang feng tong sheng san (Chinese: also may be named: Fang Feng Tong Sheng Pian (Fang Feng Tong Sheng Wan, 防风通圣片) – best names to use on Amazon) or Bofu-tsusho-san. A professional user of Microbiome Prescription reported a major increase of Akkermansia with his patients using this and forwarded two studies that illustrated this.
- Bofutsushosan improves gut barrier function with a bloom of Akkermansia muciniphila and improves glucose metabolism in mice with diet-induced obesity. 
- Increase of Akkermansia muciniphila by a Diet Containing Japanese Traditional Medicine Bofutsushosan in a Mouse Model of Non-Alcoholic Fatty Liver Disease 
So, what else does Kampo do?
- Bofutsushosan Significantly Ameliorated Betamethasone-Induced Cushing’s Syndrome 
- Mechanisms for the anti-obesity actions of bofutsushosan in high-fat diet-fed obese mice. 
- Effects of the oriental herbal medicine Bofu–tsusho–san in obesity hypertension: a multicenter, randomized, parallel-group controlled trial (ATH-D-14-01021.R2). 
For additional studies, see pub-med.
- Triggering Akkermansia with dietary polyphenols: A new weapon to combat the metabolic syndrome?  cites:
- grapes (Reservatol)
- For additional items to help (or hurt) see this page (1400 items listed).
As mentioned above, we rotate probiotics as a standard practice in our household. I am starting both the Akkermansia and Kampo and plan to do a new microbiome report in a month (so 2 months until this post gets updated). I expect a significant jump in my Akkermansia. At last test, it was 570 or 0.05% (via Biomesight), below the lower end of Dr. Jason Hawrelak Recommended ranges.
A 7 to 8 fold increase was seen after a month. The probiotic was stopped for a week before the sample was taken.
As an unintended side-effect, a lost of 7+ kilos over 7 weeks without changing diet (or dieting)!!! I have also taken high dosages of bifidobacterium infantis with it. It is known to encourage akkermansia too.