At the end of the day, political parties are there to win elections, not to retain an ideological purity. Encroach on enemy territory and pinch the ideas with broad appeal that you can also live with.
It might take a lot of hammering and sawing (and a fair amount of compromise) but it should be possible to make the pieces fit.
No more pillorying or envying or trying to pull down the rich. Instead, harness their success in the form of fair (not punitive) taxes.