Dual gear cable setup for aerobars and handelbars
Almost all derailleurs are pulled by cables. To be able to operate one derailleur from either of two shifters, or one brake from two levers, you'll need some mechanism to split the cable.

As always, while Stack Exchange usually asks us not to recommend products, I can't answer this question without mentioning some particular products. These aren't recommendations. Please do further research on your own.

Take caution that the item you need does not appear to be a common component, and there doesn't appear to be standard terminology. That said, this item by Jtek will do exactly what you ask. It appears to work with either derailleur or brake cables. They call it a cable splitter. I have never tried one, so you may need to buy at your own risk. You would need to purchase an STI shifter for your base bars (i.e. the bullhorn bars).

Just by looking at the pictures, I don't think the system will have a huge increase in cable drag, which is what I was worried about. The bigger worry may be that the system is depicted with older Shimano levers that route the derailleur cables externally. Since the late 2000s, Shimano have routed the cables under the bar tape, as SRAM and Campagnolo. With that orientation, you may need to mount the cable splitter further back so that you can route cables straight in from both the base bar shifter and the bar end shifter (i.e. the one on your aero bar end) into the splitter. You may need to experiment with the setup. I would suggest calling Jtek.

Note that the term cable splitter can refer to items that help you disconnect and reconnect cables on S&amp;S or Ritchey travel bikes, where the bike frame itself breaks into two to pack. You don't want this sort of cable splitter unless you have one of these travel bikes. I have a feeling this usage is the more common one, so look at what you Google.

The term cable doubler can refer to systems which enable you to pull one lever to actuate both brakes simultaneously. Problem Solvers makes a brake cable doubler that will do this, or will enable you to actuate one brake from either of two brakes. I haven't investigated thoroughly, but some may use this term to refer to the component you need.

Last, answering for completeness: you could get electronic shifting. It is expensive. It's not objectively worth it for most cyclists. When Dura Ace Di2 came out, many people derided it as an answer looking for a question. They had a point. But first, it really was quite a good answer! Second, and more relevant to you, it does in fact enable you to mount satellite shifting buttons in multiple locations on your handlebars.

This answer has implications for adaptive cycling. If you have a hand amputated, or if you've lost fingers on one hand, you'll seriously benefit from some type of adaptation enabling you to shift or brake from one shifter. Even for able bodied drop bar cyclists, it can help to be able to shift from multiple locations when crossing rough terrain.