I'm curious if there's many other examples of the US Postal Service collaborating with another country's postal service to make something like this. It's always neat to see a US postmark on foreign material.
The USA has done joint stamp issues many time. The largest which immediately comes to mind is the 1992 Columbian set issued as a joint issue of the USA, Italy, Portugal, and Spain; largest in number of joint stamps and countries involved.
Japan is know for some wonderful papers but the envelope you show was likely made by a cachet maker and not the postal agency.
Now to your cover:
This is not a joint issue. The USA stamp was honoring the US/Japan treaty ratified in 1860. However, the treaty interested some Japanese philatelist likely due to the travel by Crown Price and wife and your cover was born.
The combination of related or even unrelated stamps on combination
FDCs is quite common. Google or search on eBay Hideaki Nakano.
Here, as noted, the added Japanese stamps were issued in 1959, April 10th. As to the 27th date, mistakes happen or dates are changed.