  The pair of real reductive groups
$(G,H)=(\operatorname{GL}(n+1,\mathbb{R}),\operatorname{GL}(n,\mathbb{R}))$ is
a strong Gelfand pair, i.e. the multiplicities
$\dim\operatorname{Hom}_H(\pi|_H,\tau)$ are either $0$ or $1$ for all
irreducible Casselman-Wallach representations $\pi$ of $G$ and $\tau$ of $H$.
This paper is concerned with the construction of explicit intertwining
operators in $\operatorname{Hom}_H(\pi|_H,\tau)$, so-called symmetry breaking
operators, in the case where both $\pi$ and $\tau$ are principal series
representations. Such operators come in families that depend meromorphically on
the induction parameters, and we show how to normalize them in order to make
the parameter dependence holomorphic. This is done by establishing explicit
Bernstein-Sato identities for their distribution kernels as well as explicit
functional identities for the composition of symmetry breaking operators with
standard Knapp-Stein intertwining operators for $G$ and $H$. We also show that
the obtained normalization is optimal and identify a subset of parameters for
which the family of operators vanishes. Finally, we relate the operators to the
local archimedean Rankin-Selberg integrals and use this relation to evaluate
them on the spherical vectors.
