So long as you don't use the services of US or UA on the ticket at all ( and of course aren't a US citizen ) you could for example fly as follows on a Star RTW - just remember that you must make a minimum of three stops

DCA AC x/YYZ AC HAV AC YYZ AC LON NH TYO AC x/YYZ AC DCA

or anyone of a number of other routings depending on where else you might want to go

have fun - Cuba is somewhere I have always wanted to go - preferably while the Americans still have their daft embargo on it so that it is not overrun