Presentists think that the past and future are unreal but the present is real. I was going to do a tongue-in-cheek post about an opposed view where we have the past and future but no present. But as I thought about it, the position grew a little on me philosophically, at some expense of the tongueincheekness. Still, please take all I say below in good fun. If you get a plausible philosophical view out of it, that's great, but it's really just an exercise in philosophical imagination.
One way to think about antipresentism is to imagine the eternalist's four-dimensional universe, but then to remove one slice from it. Thus, we might have 1:59 pm and 2:01 pm, but no 2:00 pm. Put that way, the view isn't particularly attractive. Still, I do wonder why it would be more unattractive to remove just one time slice than to remove everything but that one time slice as the presentist does. It would, of course, be weird for the antipresentist to say that events first exist in the future, then pop out of existence just as one would have thought that they would come to be present, and then pop back into existence in the past. But perhaps no weirder than events coming out of nothing and going back into nothing, as on presentism. This way to think about antipresentism makes it a species of the A-theory.
But the antipresentisms I want to think about are ones that might be compatible with the B-theory. Start with the famous puzzles of Zeno and Augustine about the now. Augustine worried about the infinite thinness of the now. Zeno on the other hand worried about the fact that there are no processes in the now; there is no change in the now since within a single moment all is still.
One way of taking these ideas seriously is to see the present as an imaginary dividing line between the past and the future. There is in fact no dividing line: there is just the past and the future. (I think Joseph Diekemper's work inspired this thought.)
We might, for instance, instead of thinking of times as instants think of the basic entities as temporally extended events or time intervals, not made out of instantaneous events or moments. An event or interval might be past, or it might be future, or—like the writing of this post—it might be both past and future. (Thus, "past" and "future" is taken weakly: "at least partly past" and "at least partly future".) Some events or time intervals have the special property of being both past and future. We can stipulate that those events or time intervals are present. But they aren't real because they are present. They're just lucky enough to have two holds on reality: they are past and they are present. (In this framework, the presentist's claim that only present events are real sounds very strange. For why should reality require both pastness and futurity—why wouldn't one be enough?) There are no events or time intervals that are solely present.
There is a natural weakly-earlier-than relation e on events. If we had instants of time, we would say that EeF if and only if some time at which E happens is earlier than some time at which F happens. But that's just to aid intuition. Because there are noever instantaneous events, every event is weakly earlier than itself: e is reflexive. It is not transitive, however. The antipresentist theory I am sketching takes e to be primitive. There is also a symmetric temporal overlap relation o that can be defined in terms of e: EoF if and only if EeF and FeE.
If we like, we can now introduce abstract times. Maybe we can say that an abstract time is a maximally pairwise overlapping set of time intervals (or of events, if we prefer). We can say that t1 is earlier than t2 provided that some element of t1 is strictly earlier than some element of t2 (where E is strictly earlier than F provided EeF but not FeE). I haven't checked what formal properties this satisfies—I need to get ready for class now (!).