Probably neither and definitely not Virginia bluebells. I would guess Hyacinthoides hispanica or a hybrid of that. English bluebells are rarely sold. I purchased what was labeled that a few years ago but when they flowered, they were not Hyacinthoides non-scripta. The bulb supplier agreed it was not as advertised and sent me some other types of bulbs for their error.

I did a Google search for Virginia bluebell roots and clicked the ‘images’ button and got this There are quite a few pictures there that look like yours (if the link doesn’t work, try to do your own Google image search...) ACK! The link didn't appear at all! Try a Google search....

WCHopeful - I'm sorry you are struggling with this and that your plans have been thwarted. I do not have this plant on my property so I can't contribute anything specific. But I have had invasive plants that have resisted all efforts to be free of them. One was witch grass, that I really did give up on and gave up my 25x25ft vegetable garden when they overran it. I had tried to pull them, but every time you left a tiny piece of root, they multiplied. Then I made the mistake of using an electric tiller and tried just tilling over the whole 25ft garden area. Big mistake, all it did was multiply them even more. So I stopped growing anything there for 2 full years. But I always was on the lookout for a solution. I was in a forum reading threads and someone was talking about smothering an invasive plant. And I got the idea of taking a very large heavy plastic sheeting we had leftover from an ice rink we made for the kids when they were little. We covered the whole area and left it there for a year. It was a hot dry summer and that area baked. After we removed it a year later, I haven't seen that witch grass back since. I have another invasive that is in different areas in my garden and it just keeps coming back. It was all around the base of a rose I enjoy and it was just increasing and encroaching on it, so as a last resort - I am an organic gardener and won't use a chemical - and we had repeatedly pulled it like 20x in one growing season to no avail. This time I dug the whole area out soil and all, about a foot deep and threw away all the soil that came out of it. I was able to leave most of the rose roots covered well, but I left the foot deep trench open all winter. So far this spring I have not seen any. And it has come back in other areas, so I know it already should be growing. So I filled it in and so happy to see it gone. BUT - we'll see....lol. Whether it will be back before the growing season is over. You could try this method in a small area, by digging out all the bulbs and the soil around one of your favorite plants in the autumn and leave the trench open and fill it in with new soil in the spring, if the plants don't come back. If that worked, then you could just do a little bit at a time at your own pace. I would like to ask, is this bulb reproducing by seed? Is that why you removed all the sod and soil and it still came back? And do you have another area on your property that you can have another garden space? If you had it running wild on a large property away from where you were trying to garden and just left it, I could see doing that. But if it is overtaking valuable space that makes it impossible to use, I guess I'd think about trying some different ideas and keep looking for a solution. You could experiment with different methods on small patches so you could see if any of them were effective. One other thing you might try if your preference is to just leave it, is to plant something that would fight with your hyacinths and keep it from increasing further. something that is also invasive? It is running over more delicate plants like ferns, etc. I wonder how you would feel about adding Autumn Fern to the mix, which also spreads?

Well if both are already happy in your garden I don’t see why you shouldn’t plant them together. The bluebell bed would look much better with something growing in it. I can’t say anything about watering because I've no experience with any kind of irrigation. But I’m surprised it would bother daffodils. They are fine with Spanish bluebells here, apart from the bluebells being space hogs. Your Begonias are probably B grandis subsp. evansiana.