  We call an edge-colored graph rainbow if all of its edges receive distinct
colors. An edge-colored graph $\Gamma$ is called $H$-rainbow saturated if
$\Gamma$ does not contain a rainbow copy of $H$ and adding an edge of any color
to $\Gamma$ creates a rainbow copy of $H$. The rainbow saturation number
$sat(n,{R}(H))$ is the minimum number of edges in an $n$-vertex $H$-rainbow
saturated graph. Gir\~{a}o, Lewis, and Popielarz conjectured that
$sat(n,{R}(K_r))=2(r-2)n+O(1)$ for fixed $r\geq 3$. Disproving this conjecture,
we establish that for every $r\geq 3$, there exists a constant $\alpha_r$ such
that $$r + \Omega\left(r^{1/3}\right) \le \alpha_r \le r + r^{1/2} \qquad
\text{and} \qquad sat(n,{R}(K_r)) = \alpha_r n + O(1).$$ Recently, Behague,
Johnston, Letzter, Morrison, and Ogden independently gave a slightly weaker
upper bound which was sufficient to disprove the conjecture. They also
introduced the weak rainbow saturation number, and asked whether this is equal
to the rainbow saturation number of $K_r$, since the standard weak saturation
number of complete graphs equals the standard saturation number. Surprisingly,
our lower bound separates the rainbow saturation number from the weak rainbow
saturation number, answering this question in the negative. The existence of
the constant $\alpha_r$ resolves another of their questions in the affirmative
for complete graphs. Furthermore, we show that the conjecture of Gir\~{a}o,
Lewis, and Popielarz is true if we have an additional assumption that the
edge-colored $K_r$-rainbow saturated graph must be rainbow. As an ingredient of
the proof, we study graphs which are $K_r$-saturated with respect to the
operation of deleting one edge and adding two edges.
