Is it compatibale with socialism? Of course. But thats a very wide ranging "term". Technically New Labour are Democatic Socialist.
Compatibal with marxism - of course not.
In your hypothetical post communist revolution - no, private property will be abolished (along with parties ;P).
Captain Blackadder: You see, Baldrick, in order to prevent war two great super-armies developed. Us, the Russians and the French on one side, Germany and Austro-Hungary on the other. The idea being that each army would act as the other's deterrent. That way, there could never be a war.
Private Baldrick: Except, this is sort of a war, isn't it?
Captain Blackadder: That's right. There was one tiny flaw in the plan.
Lieutenant George: O, what was that?
Captain Blackadder: It was bollocks.