Subdirect irreducibility is a central notion in universal algebra. Trying to formulate it as abstractly as possible, your object is subdirectly irreducible if, in the opposite category, it does not have any nontrivial cover by regular subobjects. That is, in a sufficiently (co?)regular category $\mathbf C$, an object $X$ is subdirectly irreducible if, in the opposite category, for any family $X_i\rightarrowtail X$ of regular subobjects with the induced map $\coprod_iX_i\to X$ (regular?) epi, one of the $X_i$ must be $X$ itself.

Thus an "universal algebra-like" category is such that in its opposite, every object is covered by subobjects which are subdirectly irreducible in the above sense.

While this looks (very!) familiar, to my perplexion I cannot really come up with a specific interesting examples of categories like this which would not come from universal algebra. I have vague feeling this must have to do with local compactness and such, but I am blocked by the fact that subdirect irreducibility of finite unions of subdirect irreducibles is still another condition which I don't know where to place.

Do you have any interesting examples of this kind, from whatever field?