Smoothness of $\frac12[W_0(x)+W_{-1}(x)]$ for real $x<0$ The Lambert W-function, i.e. the multivalued inverse of $z=we^w$, has countably many complex-valued branches $W_k(z)$. The relations between the branches are a bit involved and are summarized here. 
We will consider the behavior of the $k=0,-1$ branches for $x<0$. Using Mathematica, we obtain the following plots of $W_k(x)$ along the negative real axis:

Setting aside the green line for the moment, the two plots give the real and imaginary parts respectively of $W_0(x)$ (blue) and $W_{-1}(x)$ (orange). From this, we see that both branches are real for $x\in (-1/e,0)$. (The $k=0$ branch is additionally real for positive real $x$; no other branches obtain real values along the real line.) This is not surprising, as these branches correspond to the two real-valued inverses of $z=we^w$ along the real line.
What's perhaps surprising, though, is that (according to this plot) $\overline{W_{-1}(x)}=W_{0}(x)$ for all $x\in (-\infty,-1/e)$. (A derivation of this fact may be found in the Q&A linked in the comments below.) From this, we conclude that the average of these two branches, $\frac{1}{2}(W_0(x)+W{-1}(x))$, is real for all negative real $x$. This is the green line plotted above, and from the first plot we further ascertain that $\frac12 (W_0+W_{-1})$ is smooth across the point $x=-1/e$. (Further plotting in Mathematica suggests that $\frac12 (W_0+W_{-1})$ is analytic for all $x\neq 0$ such that $-\pi < \text{arg }x \leq \pi$.)
for all $x<0$ but not holomorphic across the real line.) By contrast, the two branches have square-root branching at $x=-1/e$.
This last property of $\frac{1}{2}(W_0+W_{-1})$ remains mysterious to me, so my question is:

Why is $\frac{1}{2}(W_0(x)+W{-1}(x))$ a smooth function for all real $x<0$?


It’s not a full answer, I only want to show why $\,\displaystyle\frac{W_0(x)+W_{-1}(x)}{2}\,$ is real for $\,x<0\,$ .
We can parameterize $\,\displaystyle W_0(x)=-\ln ((1+\frac{1}{t})^t) \,$ and $\,\displaystyle W_{-1}(x)=-\ln ((1+\frac{1}{t})^{t+1})\,$ 
(where $\,t\,$ is complex) and it follows $\,\displaystyle
\frac{W_0(x)+W_{-1}(x)}{2}=-\ln ((1+\frac{1}{t})^{t+\frac{1}{2}})\,$ .
Using e.g. $\,\displaystyle t:=\frac{1}{e^{-i\alpha}-1}\,$ with real $\,\alpha\,$ we can express the complex line of $\,W_0(x)\,$ by $\,\displaystyle \frac{i\alpha}{e^{-i\alpha}-1}\,$ 
and $\,W_{-1}(x)\,$ by $\,\displaystyle \frac{-i\alpha}{e^{i\alpha}-1}\,$ so that we see $\,\overline{W_{-1}(x)}=W_0(x)\,$ . 
It follows $\,\displaystyle
\frac{W_0(x)+W_{-1}(x)}{2}=-\ln ((1+\frac{1}{t})^{t+\frac{1}{2}})= 
-\frac{\alpha}{2}\cot\frac{\alpha}{2}\,$ which is real (for $\,x<0\,$). 
