The first person we'd have to convince of this in Wittgenstein himself, who took a somewhat dismissive, self-deprecating tone towards his earlier work in his later work (see the oblique self-reference in Philosophical Investigations §23).
I think it's more appropriate to see this as an evolution in Wittgenstein's thinking, not (1) mutually inconsistent theories or (2) independent but supplementary theories. Up and through the Tractatus Logico-Philosophicus Wittgenstein was fully aligned with the intent of (Russell's) Analytic Philosophy. He wanted to define philosophy as the application of a rigorous mathematized philosophy: something where we could translate freely between logical propositions and tangible real-world phenomena without getting lost in subjective or metaphysical maundering. As he says at the end to the Tractatus:
The right method of philosophy would be this. To say nothing except
what can be said, i.e. the propositions of natural science, i.e.
something that has nothing to do with philosophy: and then always,
when someone else wished to say something metaphysical, to
demonstrate to him that he had given no meaning to certain signs in
his propositions. This method would be unsatisfying to the other — he
would not have the feeling that we were teaching him philosophy — but it
would be the only strictly correct method.
In short, here he wants to maneuver around anything that cannot be reduced to a 'natural sciences' type propositions by claiming that one of the 'signs' (terms, words) has no meaning (no clear and observable referent).
However, Russell's version of Analytic Philosophy ran into significant difficulties, mainly around the problem of 'denoting': of defending the unambiguous reference between a word (sign) and a unique real-world phenomenon. See Frege's "On Sense and Reference" and Russell's "On Denoting". While Russell et al were focused on trying to shore up the definitiveness of referencing, Wittgenstein went the other direction and began wondering whether he could rebuild what he wrote in the Tractatus using references that were purely conventional: determined only and entirely by use in language. (see PI §2, and the subsequent discussion). We can even see the beginnings of this line of thought in the Tractaus itself (TLP §§3.33-3.334), where he makes Russel's Paradox disappear by asserting that:
No proposition can say anything about itself, because the propositional sign cannot be contained in itself
...effectively distinguishing between two linguistic states that he would later come to call language games.
It seems clear to me that the 'early' and 'late' Wittgenstein are merely Wittgenstein developing his philosophy over time. People still working in AP or its descendants (e.g., Popper's school) tend to like the Tractatus, but to dismiss the Philosophical Investigations as an incomprehensible curiosity; they are the source of the 'two Wittgensteins' ideation.