Do you have any pics of the flowers, or can you describe them, color, shape, number of petals?

Thanks. That's helpful. I've never heard of remnants of blooms that look like seed pods, but maybe they are just the base of the flower once the petals have fallen off? If so, I guess I don't have seeds. Here is a picture of the flowers, some of which are still present on the plants. These are exclusively container plants. Yes, I suppose they'd be kind of invasive.

I should add that when I pull off the flowers and roll them around in my fingers, there ARE little balls inside. There are tiny brown protrusions near the base of the green stems. Not sure I'd call them "thorns", though. Are these significant?

Like the vegetable Asparagus, Asparagus ’fern’ is dioecious. If yours is a male plant it will not make berries.

AF have separate male and female plants. If you only have the one plant, non pollinated flowers makes sense.

Ah, that must be the reason. A male plant. Thank you! Now, I had always thought that Asparagus Fern was a funny name, since it didn't look like it had anything to do with asparagus. But, in fact, it is genus asparagus! That being the case, it also isn't really a fern. Just looks like one.

It could be female and have unfertilized ovaries, like Tiffany said. I was googling male flowers and what happens to them after blooming, and I couldn't find any pictures. They probably just fall off? Do you want the species identified, or the brown 'seeds' identified?

If you end up with seeds being produced, it won't be just a container plant.

Looks like Asparagus densiflorus Sprengeri Group, a pretty common and easy care houseplant. I grew one for many years but couldn't stand the smell of the flowers when they bloomed so eventually got rid of it.

If a male plant or unpollinated female, you can purchase seeds but is very easily propagated by dividing the tubers in spring.

I was wrong about it being female. The flowers in your pic are male. They have the orange anthers. Female flowers have a green ovary with no yellow or orange parts. I don't think they flower outside here in the midwest, so I'm not familiar with their flowers, or what happens to them after blooming. They are used for container plants, with Pelargoniums and purple spikes,(Cordyline, Dracaena),Vinca vines and other annuals.

The Cordyline is C. australis. With 'Spikes' apparently being a made up marketing name having no basis. (Unless somebody thinks they have selected a uniform seed strain conforming to the definition of a cultivar). And Dracaena indivisa being a synonym. With there being associated confusion with C. indivisa, a different, much less frequently cultivated species also having a limited wild presence. In contrast to C. australis which is common in nature.