What should the count rate be?
The theoretical rate of cosmic rays is of the order of 1 count per minute per cm2 of active area, but it does depend on solid angle, so this number is only approximate. Assuming each GM tube has a broadside active area about 10 cm2, the number of counts maybe 10 per minute.
If the rate of each tube; S1, S2, and S3 counts per second and the coincidence gate width is τ seconds, then if counter #1 is ON S1τ , and the random coincidences in tube #2 at a rate R12 = S1S2τ random coincidences per second. So the random coincidence rate should be less than 1% of what is expected for the real coincidence rate, or about 0.1 counts per minute for a 10 cm2 Detector. (Many thanks to Bob S for this information)
One of the reasons for false counts in a Geiger–Müller array detector maybe due to Compton Scattering, where an interaction between charged electrons within the detector and high energy photons result in the electron being given part of the energy, causing a recoil effect of a high energy photon into the adjacent detector causing a false coincidence detection. In other words this causes cross-talk interference between GM Tubes
Consequently radiation shielding is required between each GM Tube of either 6mm of lead, 12mm of copper or 25mm of aluminium (note Iron is unsuitable). (Many thanks to Bob S for this information)
August 13th 2009 - To date tests carried out using radiation shielding between the GM tubes don't indicate that this is a real problem, however any cross-talk between each the tubes or the electronic is a real concern and should be design into any detector array.
Detector pulse width
In theory the Geiger–Müller tube is a very good detector of Muons (Cosmic Rays) however it would seem that filtering out background radiation using a simple coincidence detector is problematic due to the Geiger–Müller tube response and decay time (Pulse Width) when an ionising particle has been detected.
Consequently, the wider the Pulse Width the greater the number of false positives. The means a pulse shorting or quenching circuit is also needed to shorten the Pulse Width to a period closer to the expected flight time of the Muon between tubes, but not too narrow that the electronics cannot measure relative coincidence. Some improvement might also be achieved by spacing the tubes further apart, but this also has the negative effect of decreasing the aperture of the detector.
Although coincidence implies simultaneously, in reality we are talking about almost simultaneous, this is because a muon created by a cosmic event is travelling at near the speed of light 0.998c, so if the detectors are only spaced 2.5cm apart the actual flight time of a muon would only be 0.08ns. However as the detector and electronics response and delay times are slower than this, we can say in "real-life terms" it is simultaneous.