Or maybe both? Quoting from Pierced for Our Transgressions: Rediscovering the Glory of Penal Substitution by Steve Jeffery, Michael Ovey, Andrew Sach:

[S]ome have argued that God does not actively punish sin, but instead allows sinners to reap the ‘natural’ consequences of their actions. The idea here is that God does not intentionally impose these consequences; they are the natural outcome of events, a moral cause and effect analogous to physical cause and effect. Thus the husband who commits adultery suffers the consequence of collapsed trust in his marriage, and the man with a violent temper experiences the inevitable decay and loss of friendship. Within such a framework, it is claimed, we should not talk about God intentionally or actively punishing sin.

But this is a mistake, because it downplays an aspect of God’s work in creation; namely, his continuing, intentional sustaining of all things. All ‘natural’ consequences, whether physical or moral, take place because God continues to will the existence of the universe and the causal relationships that occur within it. Since the moral consequences of sin are willed in this way, they have the character of divine punishment.

Of course, so-called ‘natural’ consequences of sin are not the only kind of divine punishment there is, but it is real divine punishment.