I'm considering the following treatment:
After 1X-Pass-1Y-Double, pass by opener says:
- All doubles by the partnership are now penalty
- Responder is to double two of a minor bid with 4+ length in the minor, even if minimum
- Responder is to double 2 of a major that doubler showed, with Hxx or better in the suit and 11+ points (or less points, better suit)
- Responder is to double 3 of any suit with Hx or better and 11+ points (or less points, better suit)
- 3NT by responder directly over 3 of a suit shows xx in that suit, 12+ points, no good bid.
- Opponents cannot play undoubled below 2H
- This double does not promise extra points, just puts the partnership in headhunting mode
Since the opponents cannot play undoubled below 2H, if it goes 1X-Pass-1Y-Double--Pass-Bid below 2H, responder's pass here will be forcing, since opener must bid or double.
If opener does not want to go into headhunting mode, perhaps because the opening hand is distributional and/or quite weak, then opener makes the normal descriptive call over the double. This will include support redoubles if the partnership plays them.
The reason we don't play a headhunting redouble is this would allow the next hand to pass and get the doubler into action to find the best fit. Instead by opener passing, the next hand has to act unilaterally, and might land in an awful spot.
If it goes 1X-Pass-1Y-Double--Pass-Pass, responder passes to play there, can bid naturally, non-forcing (jumps are invites), or can redouble which asks opener to describe hand as if there was no double, and then the bidding continues as if there was no double. Since responder can pass to play in 1Y doubled, if opener does not show support then any natural bid of Y by responder is not to play but asks for a Y stopper.