German public health insurances are often "Körperschaften des öffentlichen Rechts". While this means that they're "entities under public law", they're typicall

German public health insurances are often "Körperschaften des öffentlichen Rechts". While this means that they're "entities under public law", they're typically running on their own, as a type of non-profit company, with board of directors, annual reports, ... Probably the closest US organization type would be a non-profit under 501(c)4.