How can the effect of a pulse sequence on a strongly coupled spin system be analysed?
The product operator formalism still applies to strongly coupled systems.  The simple "magic formula" for calculating a closed-form solution to the step-wise time evolution only applies when the terms of the Hamiltonian commute with each other, which is not the case in a strongly-coupled system.  Maybe more accurately stated: in a weakly-coupled system, the non-commuting terms can be neglected.  The other assumption that allows the removal of non-commuting terms is that the pulses are very short compared to other evolution times, so during pulse periods the free-evolution terms are omitted from the Hamiltonian.

The effects of a pulse sequence without these on such a system can still be found by computing the exponential of the matrix rather than using the shortcut.

Now a shameless plug: the SpinDrops tool (http://spindrops.org) can be used to simulate and visualize the effects of pulse sequences with arbitrary Hamiltonians. It shows an exact 1-1 mapping of coupled-spin operators to the visualization which can not be done with the vector model.  The setting 'Pure Pulses' can be disabled to include the free-evolution Hamiltonian during pulses, but incorporating strong coupling would currently need to be done by defining the Hamiltonian explicitly.