Can we describe quantum mechanics without the non-physical aspects of their complex wavefunction formalisms?

This question is a little strange and a bit hard to articulate, so please read carefully.

In the wavefunction formalism, we find lots of terms with complex phase factors, like e^-iaEt, which is probably the most important one to discuss. Specifically, because of two things- 1) E can be set arbitrarily through where we choose to set out zero point, thus changing the rate of change of the complex phase. 2) Given any two wavefunctions (say we're summing them), we only ever care about relative phase and relative rate of change of phase, whatever specific values we choose can be shifted through a change in the definition of [;E;] or a redefinition of [;t=0;].

Essentially when we use these complex number formalisms, most (but not all) of their complex aspects are aphysical, they can be changed arbitrarily.

So my question I guess is, is there a way of doing things without this artifice? Purely in terms of physically meaningful quantities which tell us how wavefunctions (or to be more general, quantum states) evolve, interact, and the measurables they produce?

I'm not necessarily too concerned with being super practical for the majority of day-to-day calculations, this is in essence a more philosophical concern. Just like how I like that all of physics can be represented in completely co-ordinate free ways even as most calculations I'd do involve me choosing a basis to work in for convenience, I'd be happy to know about and talk about quantum mechanics without the aphysical parts even as I use them for convenience.

And please, before you dump a bunch of highly technical maths, recognise that a lot of what seems self evident to you may not be to others, so please explain the reasoning and motivations for definitions/formulations etc when you introduce them.