I saw this old closed question on Physics.SE and thought it would be perfect for here. The original author is long gone, so I thought I'll just have to post it myself.
We can consider factors of plot and literature, not just hard science. But I do intend for it to be hard-SF.
In Greg Egan's The Arrows of Time he had a crew visit a region where the thermodynac arrow of time was in the opposite direction. The ambient sunlight was not visible to them and they would unmake footprints.
But I think that's not right. Consider what would happen if you prepared such a region by somehow reversing the momentum of every particle. It is very precarious with every gas molecule having to be exactly right! As soon as you observe it, you peturb that delecate state and time runs normally from that point forward.
Consider an experiment where a gas cloud in a large isolated chamber will rush into a bottle, because the state of every molecule is just so. If you go into the chamber, it won't work anymore because you mess up those perfect trajectories. (Unless the trajectories were chosen with knowledge of your exact effect in mind...!)
Suppose you had a wormhole that led to another part of the universe where time ran backwards or the orientation of the exit mouth were such that it dropped you into the time dimension facing the wrong way.
You don't experience the environment to be anti-matter: as in Egan's story it would be anti-matter if your time arrows were aligned, so you have matching matter polarity but opposite entropy gradiants and (whatever that means if it's more than just entropy) opposite time directions.
What happens? How?
Maybe you are like the gas bottle cloud and the large environment at your destination peturbs you. Does time do funny things? Do time and entropy break up into separate concepts? Could you go down in an elevator that's ascending, escaping in the same ride that the police use to arrive at the scene of the crime?