Abs and RealAbs have different derivatives. Abs is meant to work in the complex domain, where it is not differentiable. RealAbs is differentiable in the real domain. Compare:

D[Abs[x], x]
D[RealAbs[x], x]
% // PiecewiseExpand

I wish there was a RealIntegrate that behaved this way:

RealIntegrate[1/x, x]

Log[RealAbs[x]]

It might not be completely impossible to implement. Look up $PreRead in the documentation. Learning how to use that to implement your idea would take some time, study, experimenting and almost certainly some failures. Implementing your idea might break some parts of Mathematica, perhaps in surprising and possibly even impressive ways, and break some people's programs. But that doesn't mean it is impossible to do. Just be careful and don't get yourself or anyone else into trouble.

Oh I didn't know about this. Thanks!

It still would be a cool feature, but I guess this makes it impossible to implement.