ShadowNumber for iPhone and Android. The app is coming soon to the stores.
Works on iPhone and Android, with the SIM already in your phone.
You can activate it on the website now, and use it in the app when it arrives.
Calls to the number ring on your phone's regular line in cellular quality, and outgoing calls go out through your phone's own dialer. You don't need internet to talk, and any phone can call the number.
Real text messages: any phone can text the number, and you can text any phone in Israel from it. Not an in-app chat.
Calling over Wi‑Fi is just an option, for when you have no reception.