**A few preliminary remarks:**

1) Complexifications of Banach lattices are in fact a special case of the more general concept of *complexifications of real Banach spaces*.

2) Most books and articles about complex Banach lattices which contain spectral theoretic results focus on *positive* operators (or, say, generators of positive semigroups) which are, of course, real. 

3) One reason for 2) might be that real operators can actually defined on every complexification of a real Banach space (no matter whether a lattice structure is present), so the question about "spectral properties of real operators" belongs to the realm of complexifications of real Banach spaces rather than to the realm of complex Banach lattices. 

4) While most books on Banach lattices contain a chapter on complex Banach lattices (which is, unfortunately, very brief in most cases), there does not seem to exist much literature on complexifications of real Banach spaces (although the topic often seems to be considered as some kind of "standard folklore" among functional analysts).

The only article I'm aware of which contains a rather extensive treatment of complexifications of real Banach spaces is "G. A. Muñoz, Y. Sarantopoulos, and A. Tonge: Complexifications of real Banach spaces, polynomials and multilinear maps. Studia Math., 134(1):1–33, 1999." Unfortunately, though, spectral theory is not treated there.

**References to learn about spectral properties of real operators:**

If you primarily want to learn about the spectral properties of real operators, you can find some information in the following references (I apologize for the self-advertisement, but these are really the only references I know for this topic):

[1] Appendix C in my [PhD thesis](https://dx.doi.org/10.18725/OPARU-4238) contains a thorough treatment of complexifications. Spectral theoretic aspects are discussed in Section C.3, and a few spectral theoretic properties of real operators are discussed in Proposition C.3.2 (the fact that the operators considered in this proposition are exactly the class of real operators follows from Proposition C.1.6). However, I should add that Appendix C contains almost no proofs, for the following reason (which reflects, of course, only my personal point of view): when it comes to complexfications, it is in most cases more difficult to find the correct statements rather than to prove them (proving them is very straightforward in most cases).

[2] You can also find similar information in Appendix A of the [arXiv version 1](https://arxiv.org/abs/1410.2502v1) of my paper "Spectral and Asymptotic Properties of Contractive Semigroups on Non-Hilbert Spaces" (but in version 2, which was eventually published, I removed this appendix and replaced it with a very brief discussion of complexifications at the beginning of the paper in response to a request in a referee's report).

**References to quote:**

- The above references [1] and [2] are certainly not very well-suited in case that you need a reference to quote in a paper. Indeed, [2] is only a preprint which has eventually been published without the information relevant here, and [1] contains almost no proofs (moreover, one might also question whether it is a particularly good idea to refer to the appendix of a PhD thesis for information about a quite classical topic in functional analysis - even though this topic does not seem to be particularly well represented in the literature).

- Unfortunately, I do not know any other references about the spectral theory of real operators (and I have severe doubts whether such a reference exists).

- So my suggestion is: If you need spectral properties of real operators in a paper, simply state them either in the introduction or in an appendix of the paper. You may include proofs if you feel they are necessary; but in many cases those properties are such elementary that you probably won't need to include proofs.

**Additional remark (biased by my personal opinion):**

I think it is reasonable to conlcude that a survey article about complexifications of real Banach spaces which complements the aspects discussed in the article of Muñoz, Sarantopoulos and Tonge would be a very useful source of reference. Now, all that is needed is a volunteer to write it and a journal willing to publish it...