For real $x<0$ the Bose-Einstein integral of order $s$ is given at https://dlmf.nist.gov/25.12.E15 as
$$G_{s}(x)=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{t^{s}}{e^{t-x} -1}\mathrm{d}t$$
which can be written as a polylogarithm $G_s(x) = \mathrm{Li}_{s+1}(e^x)$. For $x>0$ this value of $G_s$ becomes complex. In an old paper from 1954 (On Bose-Einstein Functions) J. Clunie defines the Bose-Einstein functions for $x>0$ by the Cauchy principal value
$$\overline{G}_{s}(x)=\frac{1}{\Gamma\left(s+1\right)}\lim_{\delta \to 0}\left(\int_{0}^{x-\delta}+ \int_{x+\delta}^{\infty}\right)\frac{t^{s}}{e^{t-x} -1}\mathrm{d}t$$ (actually the pre-factor is omitted). The table for $\overline{G}_{1/2}(x)$ with $x >0$ can be reproduced numerically with Maple as $\overline{G}_s(x) = \Re \mathrm{Li}_{3/2}(e^x)$.
Is there a known proof that
$$\overline{G}_{s}(x)=\frac{1}{\Gamma\left(s+1\right)}\lim_{\delta \to 0}\left(\int_{0}^{x-\delta}+ \int_{x+\delta}^{\infty}\right)\frac{t^{s}}{e^{t-x} -1}\mathrm{d}t = \Re \mathrm{Li}_{s+1}(e^x),\quad s>0, x>0 $$
I searched the internet without success for a relevant link, but perhaps there is a general result from complex analysis.
Edit: Thanks to R.J. Mathar's links I have the following argument: Ng cites Dingle and gives the relation to the Fermi-Dirac functions (see https://dlmf.nist.gov/25.12.E14) $G_s(x)=-\Re F_s(x+i\pi).$ If we use the integrals in a heuristic way this gives for $x>0$
$$G_s(x) = -\Re F_s(x+i\pi) \stackrel{?}{=} -\Re \left(\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{t^{s}}{e^{t-x-i\pi}+1}\mathrm{d}t\right) = -\Re \left(\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{t^{s}}{1-e^{t-x}}\mathrm{d}t\right)=\Re \mathrm{Li}_{s+1}(e^x)$$
Is the second step justified? From the DLMF notation, I would think that the argument of $F_s(x)$ should be real in the integral form, otherwise they use $z$ for complex variables?!
This might basically be covered by the article by Ng, Devine and Tooper in http://dx.doi.org/10.1090/S0025-5718-1969-0247739-X, Mathematics of computation 23 (1969) 639. Perhaps some combination of Robinson's paper http://dx.doi.org/10.1103/PhysRev.83.678 and Lee's paper on polylogarithms http://dx.doi.org/10.1103/PhysRevE.56.3909 also provide similar results.