Why is there no “real analytic continuation”

A singularity in a real analytic function disconnects the real line. Consequently, the parts on either side of the singularity need not agree anywhere. For instance, consider $\int_1^x 1/t \,\mathrm{d}t$, which gives the logarithm. Notice that $C + \log |x|$ is a solution left of the singularity at zero for any $C \in \Bbb{R}$, so there is no unique continuation.

There are $\Bbb{C}$ functions which exhibit a similar phenomenon. $\sum_{n \geq 0} x^{2^n}$ has a natural boundary along the unit circle -- it cannot be extended outside the circle.