When the current stops fitting in one bridge, people add another. paulsimik runs a double bridge; Steve Ward's Fat coil runs eight, 13 nF in the primary, about 75 A each and 600 A in total.
There are two mechanisms and they are not equally safe.
Transformers, which is the sound one
Each bridge drives its own transformer primary, and the secondaries go in series.
Steve Ward: since the secondaries are in series, their currents must be equal; if the secondary currents are equal, the primary currents must therefore be equal.
It holds even when the bridges see different voltages, from unequal busbar resistance for instance. The currents stay equal and only the contribution to the power differs, in proportion to the voltage.
Ward again, on what the arrangement really is: essentially you are putting the inverters in series, rather than in parallel.
Parallel secondaries achieve nothing, which is worth stating because it is the first thing people try.
Why the transformer core does not saturate
This is the question almost everybody asks, and the answer is worth having because it applies far beyond this arrangement.
Steve Ward: flux is proportional to volts × time / turns. The current in the secondary is compensated by the added current in the primary, so despite operating at a high level of current, the magnetic flux in the core is the same as if there were zero secondary current. The flux is set only by the small magnetising current.
Paultesla puts the same thing shorter: a core saturates from too many volts per turn, or from a DC voltage applied for too long. Not from load current.
A split tank capacitor, which is easier and is not fool-proof
Divide the capacitor bank into sections, one per bridge. It works because the capacitors' impedance at the working frequency is significant, a few ohms, so they behave as ballast resistors the way they would when paralleling transistors.
The equi-drive arrangement is mandatory: capacitors on both sides of the primary. Otherwise you are forced to join the bridge outputs directly, which is the thing being avoided. The number of banks on each side can differ, but equal is better because that maximises the impedance.
Hydron, who built exactly this, simulated one half bridge failing and reported that the results are not pretty. He added separate current transformers on each half bridge to catch the divergence.
And the failure spreads
The lesson from a real post-mortem is that the damage is always wider than the device that failed, because plasma from it reaches the ones beside it. So: space the transistors, give each its own heatsink, and use materials that take the hit. Industrial modules are potted in silicone gel for a reason.
The department diagrams draw one bridge, because one is what most coils have and because whether it is a full bridge or a half is the question that comes first. The rest of this only starts once that one is at its limit.