Firstly, this design requires positive displacement, high pressure, forced induction. This would probably be accomplished with a layout similar to the Scuderi split cycle engine... but with a very different phase difference.
The compression piston would work almost exactly like that of a Scuderi
split cycle engine, except that it reaches top dead center while the power piston is about halfway through it's upstroke. Also, the compression piston will be about half the size of a comparable Scuderi engine's compression piston, since there will be two compression strokes for every combustion power stroke.
The power piston head would contain three valves (or sets of valves): a crossover valve, an air exhaust valve, and a steam exhaust valve.
The first stroke is the steam expansion stroke. With all the valves closed, water is injected into the cylinder. The water hits the piston, and turns to steam. The pressure of the steam against the piston produces mechanical power as the piston descends.
The second stroke is more complex; at bottom dead center, the steam exhaust valve opens. When the piston is about halfway up (coinciding with when the compression piston is approaching top dead center), the crossover valve opens (with the steam exhast valve still open). Hot, high pressure recirculated exhast gas, that was previously forced into the crossover tube, rushes out of the crossover, forcing the steam out the open exhaust valve. Before too much exhast gas goes out the steam exhaust valve, it closes.
A short time after this, after all of the recirculated exhaust that had been forced into the crossover tube has reentered the chamber, and after the desired amount of fuel/air mix has entered the chamber, the crossover valve closes. The piston continues to rise, compressing the mixture of air, fuel, and recirculated exhaust gas, until autoignition occurs (as close as possible to TDC).
The third stroke is the power stroke -- mostly. After the piston has descended enough that the pressure in the chamber is only a little above the pressure in the crossover tube, the crossover valve opens, allowing some exhaust gas to go into the crossover tube. After the desired quantity of exhaust gas has gone into the crossover tube, the crossover valve is closed. The remaining exhast gas is then expanded conventionally.
The fourth stroke is a conventional exhaust stroke. The air exhast valve opens at bottom dead center, and closes at top dead center.
The steam exhast valves would lead to a steam exhast manifold, which would go to a condensor. A certain amount of entrained air is inevitable, so an air/water seperator would be needed to vent that air to the atmosphere.