> Is there any other canonical candidate which arises? My question here is mainly about opinions or some sort of vision which explains why this or that model or object of study arises naturally.
I'm surprised that nobody mentioned the following two "canonical" candidates for "realistic mathematics":
1) ZF+AD ([the Axiom][1] of [Determinacy][2]). This proves DC (but disproves the full AC) and that all sets of reals are Lebesgue measurable; an advantage over the Solovay model (see Andreas Blass' answer above) and over $L(\Bbb R)$ is that AD is quite easy to state and can be seen to some extent as a philosophical principle. ZF+AD also has other plausible consequences, including the Continuum Hypothesis in its original formulation (every uncountable subset of the reals has cardinality continuum) and on the other hand that the continuum is not well-orderable. Descriptive set theory has a lot of "nice" results proved under AD or under a weaker axiom PD (Projective Determinacy) which does not contradict the full AC. With [recent][3] [work][4] of Woodin, the "politically correct" form ZFC+PD of the original theory ZF+AD seems to be gaining considerable prominence in set theory. AD implies the consistency of ZF (so ZF+AD is substantially stronger than ZF), but on the other hand ZF+AD is consistent relative to ZF + a large cardinal axiom (in fact, $L(\Bbb R)$ satisfies AD modulo this axiom).
2) [Positive set theories][5]. An intuition for "realistic mathematics", which also reminds of $L(\Bbb R)$, could be that all sets should have something to do with the continuum, insofar as they are not definable in pure logical terms. A more specific intuition, which also reminds of AD, could be that all sets should have something to do with a topology (or a uniform structure).
In a positive set theory, "sets" are thought of as closed subsets of some space, and non-closed subsets are the "proper classes". Discrete subsets can be identified with a model of ZFC. The complement of a set is a class (possibly proper; this is where "positive" comes from), and the closure (with respect to the topology) of every class is a set. So classes are not "bigger" than sets, somewhat unexpectedly. The Russel class $\{x\mid x\notin x\}$ is a proper class whose complement is a set, $\{x\mid x\in x\}$. I find this topological explanation of [proper classes][6] and Russel's paradox quite convincing.
For an introduction to positive set theories see survey [notes][7] by [Holmes][8], who in particular explains how to obtain a model of such a theory by collapsing a type theory. There is a variety of positive set theories; at least one of them is [equiconsistent][9] with a second-order ZFC (Kelley-Morse-Tarski theory with choice).
In ZF-like theories, sets (e.g. those in Goedel's constructive universe $L$) unfold in a
fashion that reminiscent of direct limits (=colimits), beginning with an initial object, $\emptyset$. In positive set theories, there is the universal set, and the unfolding of sets is more in the spirit of inverse limits (=limits). Here is Hinnion's description (from the chapter "Alternative Set Theories" in [this][10] book) of a model (I think it is
this model that is also known as the "$\omega$-hyperuniverse") of a positive set theory.
<img src="https://i.sstatic.net/2VMMY.jpg" width="550">
In conclusion let me cite from this [review][12] by R. Holmes of a paper by R. Hinnion:
> In this paper, the author introduces a general method of converting an arbitrary first-order structure into a uniform space, and studies the resulting notion of Cauchy completion of a first-order structure. This is a generalization of constructions used to build models of positive set theories.
The paper itself (as well as its erratum) seems not easy to find in libraries, but the construction should be similar to, if not same as, as in a previous (or rather subsequent?) [paper][13] by Hinnion.
[1]: http://en.wikipedia.org/wiki/Axiom_of_determinacy
[2]: http://en.wikipedia.org/wiki/Determinacy
[3]: http://www.ams.org/notices/200106/fea-woodin.pdf
[4]: http://www.ams.org/notices/200107/fea-woodin.pdf
[5]: http://en.wikipedia.org/wiki/Positive_set_theory
[6]: https://mathoverflow.net/questions/48305/proper-classes-and-their-consequences
[7]: http://math.boisestate.edu/~holmes/holmes/separticle.pdf
[8]: http://math.boisestate.edu/~holmes/holmes/positivenotes.ps
[9]: http://www.ams.org/mathscinet-getitem?mr=1669902
[10]: http://books.google.com/books?id=mbn35b2ghgkC&printsec=frontcover#v=onepage&q&f=false
[12]: http://www.ams.org/mathscinet-getitem?mr=MR2308602
[13]: http://onlinelibrary.wiley.com/doi/10.1002/malq.19970430404/abstract