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]]

How would you interpret |a|b|c|? As Abs[a*Abs[b]*c] or as Abs[a]*b*Abs[c]?

Have you considered using \[LeftBracketingBar] and \[RightBracketingBar]?

One more complication is that we have to deal with Abs[x] and RealAbs[x]. TraditionalForm does not make any visual distinction between the two, except in the Tooltip.

Oh I didn't know about this. Thanks!

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