Can my open source app be "stolen"?

I just finished an app for both ios and Android that I plan to make open source. But I'm worried that if I don't publish it to the App Store and Play Store first that someone else can publish it before me an potentially even charge money for it. I don't want to pay the $100 Apple fee and connect my app to my name just yet, and I still want to polish it a bit before submitting to the App Store. I doubt two of the same app would be allowed on the App Store. If someone does "steal" the app, what are my options? Can I somehow reclaim it?