Just sharing some details that I know about this topic.
There are multiple options depending on your personal goals and the level/extent of your prior training/experience with regards to foreign trained anesthesiologists doing a fellowship in USA. (This info may apply to many other specialties too)
Various options include - 1. fellowship training/going back to home country.
2. Fellowship training and staying back as faculty in a specific institution without board certification ( obviously with risks of being underpaid and restricted mobility if you want to switch institutions or jobs)
3. Fellowship training and going through ABA alternate entry path to get board certification - like a poster mentioned above. After board certification , you can possibly have equivalent mobility as someone who went through residency here in US atleast in academic settings but I would imagine it would be much more difficult in private practice settings.
4. Fellowship training and going back to residency and getting board certification - obviously fellowship training can give enough USCE for anyone to match into residency. This might be the best option if you intend to practice in US and have excellent job opportunities both in academics/PP and want the mobility to move to any state.
5. If you know you want to do residency again, going through with that as the first option instead of doing a fellowship.
6. In some rare instances, if you are a highly experienced person (typically highly qualified , well renowned in the field) , you maybe able to work as a faculty directly in US, especially at certain academic places, without any further training.
Obviously , your interests and goals might change as you go through the process and you may wanna switch from one to another - like you maybe interested in just a fellowship and going back to your country, but later switch to planning on staying in the US and may wanna do a residency here !
In addition to non acgme accredited fellowships, it is possible to do even ACGME accredited fellowships as a foreign trained anesthesiologist. But again it depends on your prior experience and the field you are interested in. Please search for "common program requirements" in the ACGME website for the specific anesthesiology fellowship you are interested in and look into the clause mentioning "exceptionally qualified international graduates" for details. That being said, as the name suggests, it would be highly competitive to get an exception to match and it is a lot of work for the programs to take someone like that in an ACGME accredited spot, but still possible if you can prove your worth! It might be virtually impossible to get into highly competitive acgme accredited fellowships ( CT, pain etc) at any place! It maybe possible to get into less competitive ones like OB/regional/peds/critical care at a decent place! Again depends on your interests.
Whether accredited or non-accredited , most if not all of the clinical programs would want you to complete USMLE steps and get ECFMG certified. Again, as people have mentioned before, be wary of some non-accredited programs where you are just a cheap labor to fill a spot. Still some people may want to do it just to get a foot in the system !
If you are interested in research fellowships, you maybe able to do it without USMLE or ECFMG certification.
It is an ever changing scenario, and you can gather more information by doing a detailed online research of ABA , ACGME and individual program websites and by emailing the programs. All the best !