Is there an iPhone app that can allow me to text an international number for free?

I know there are a lot of free texting/SMS/messaging apps out there for iPhones, but my problem is texting internationally for free.

I live in the US and have unlimited 4G and SMS, so IIm willing to use my own 4G to use the app.

However, my girlfriend lives in the UK currently (in England) and her 4G is extremely limited every month. So, I d rather her be able to use regular SMS to "text" me on this app.

Essentially, I want an SMS app that works like any other decent messaging app on 4G/wi-fi for me, but gives me a UK/England cellphone number or something so we can message throughout the day.

I just want to be able to talk to her without waiting for her to get home or worrying about whether she ll have wi-fi wherever she is. And for free on her part!

Any iPhone apps (that are tried and true) and alternate suggestions are welcome!

Added (1). I should clarify that I basically want my girlfriend to be able to use JUST her cellular service and not have to pay extra to text me back on her SMS. So if there's an app where I can get a UK phone number, that'd be awesome!

Skype. You BOTH need to have it installed on your phones.