  We perform, for the first time, a detailed long-term activity study of the
binary system $\zeta$ Ret. We use all available HARPS spectra obtained between
the years 2003 and 2016. We built a time series of the Mount Wilson $S$ index
for both stars, then we analyse these series by using Lomb-Scargle
periodograms. The components $\zeta^{1}$ Ret and $\zeta^{2}$ Ret that belong to
this binary system are physically very similar to each other and also similar
to our Sun, which makes it a remarkable system. We detect in the solar-analogue
star $\zeta^{2}$ Ret a long-term activity cycle with a period of $\sim$10 yr,
similar to the solar one ($\sim$11 yr). It is worthwhile to mention that this
object satisfies previous criteria for a flat star and for a cycling star
simultaneously. Another interesting feature of this binary system, is a high
$\sim$0.220 dex difference between the averages log($\mathrm{R}'_\mathrm{HK}$)
activity levels of both stars. Our study clearly shows that $\zeta^{1}$ Ret is
significantly more active than $\zeta^{2}$ Ret. In addition, $\zeta^{1}$ Ret
shows an erratic variability in its stellar activity. In this work, we explore
different scenarios trying to explain this rare behaviour in a pair of coeval
stars, which could help to explain the difference in this and other binary
systems. From these results, we also warn that for the development of
activity-age calibrations (which commonly use binary systems and/or stellar
clusters as calibrators) it should be taken into account the whole history of
activity available of the stars involved.
