There is an "added seconds" counter, which counts 1 second (SI
second!) every 36 seconds up to 100 per hour, while the ordinary seconds
counter is stopped.
Each full 100 "added seconds" the "subtract counter" counts 1 second (SI
seconds !), before the regular seconds counter continues.
Balancing these counters gives the length of the Mhour 3699 seconds.
In all calculations you need, mainly for navigation with speed and
distance logging, you sum up the visible(!) number of seconds: hours
times 3600 + minutes times 60 + seconds + "added counter" seconds -
"subtract counter" seconds.
All clocks on Mars in each timezone get synchronized by radiosignal from
"Airy beacon", giving MTC (Mars Time Consolidated) corresponding to UTC
on Earth.
At each full Mhour there will be 1 second dropped or kept
corresponding to the leapsecond needed, to nullify the 0.24... seconds
per sol, which are not countable in units.
In case the clock misses some beacon signal there is no problem, because
of the minor loss in accuracy per sol less than 1 second.
The total sum of seconds counted is 88776, that is 0.76 seconds more
than the correct number of 88775.24 seconds per sol.
Synchronizing the clocks on Mars with clocks on Earth is a special task
for chronometer specialists, taking into account the different span of
time needed for transmission at any time off any location! First we will
need an atomic clock on Mars, counting in Milli Milli seconds! But watch
out for Einstein's relativistic time shifting, stretching, squeezing in
space travels with different speed.
Pro
This system preserves a 24:60:60 clock using the standard second. For most civilian
purposes scheduling to the second is not necessary and most Earth digital clocks
do not display the second. Where accuracy to the second is required, Martian clock
and watch displays can be enhanced to include a counter for the additional minute
and 39 seconds added each hour. Standard Earth display of time in HH:MM:SS and
use of Earth nomenclature for time can be used, providing one can live with ignoring
the extra 99 seconds between hours.
Con
The idea that "keeping the SI second is absolutely mandatory" is a common fallacy. In fact, scientists have never used the SI second to express the time of day on Mars. Rather, they have always used the stretched Martian second (equal to 1.02749125 SI seconds). People often fail to make the distinction between the use of the SI second for deriving other physical units, and the use of the second in relation to solar angle. Since even on Mars, a circle has 360 degrees, each degree has 60 minutes, and each minute has 60 seconds, it makes no sense to have a timekeeping system other than the traditional 24:60:60 clock. The relationship between time and solar angle must be preserved. Meanwhile, conversion between the stretched Martian second and the SI second is easily done whenever it is necessary. This is what scientists have always done with regard to Mars. Why do non-scientists insist that the SI second is absolutely mandatory when scientists know that it is not?