Bridgeman writes in A Physicist's Second Reaction to Mengenlehre:"The feeling that actually existing things are not self-contradictory is so elemental as to almost constitute a definition of what we mean by self consistent. Now when we are concerned with things we are evidently concerned with some form of experience, so that we may make an even broader statement and say that experience is not self-contradictory". This appears convincing at first, but the more you think about it the less sense it makes.
Historically, many experimentally successful physical theories were inconsistent, classical electrodynamics had infinite self-interacting energies, Planck's radiation and Einstein's photoeffect theories treated light as both waves and quanta, Bohr's atom was a chimera too, quantum mechanics must have "quantum objects" interacting with distinct "classical apparatus", which nonetheless consists of those same quantum objects, and incompatibility of the standard model with general relativity famously motivated the string theory. There is no detectable 'growth' of consistency either, if anything the conciliation difficulties increase.
First, it means that the logic of scientific theories is not classical, if it were the law of explosion (contradiction implies anything) would make them useless. Scientists obviously have ways of compartmentalizing contradictions. But even trying to justify a belief in some self-consistent "theory of everything" seems problematic to me. The reality is supposed to follow some "intelligent design", Platonic or theistic, that we are discerning from experience? This is worse than what it is supposed to justify. And even if there was such a consistent "intelligent design" science is a compromise between that and human limitations. We deal only with our best approximation, and that may well be a patchwork of contradictory but practically successful mini-theories rather than a "theory of everything".
Was the logic of scientific theories studied in how it deals with internal contradictions, are there formal models of it? Are there compelling empirical or philosophical reasons for scientific theories to be (eventually) self-consistent?