Do any devices support handshaking on SPI-ish bus without extra wires
The closest thing I know for what you're asking is UART/RS-232 with RTS/CTS handshaking (4 wires total).

National Semiconductor's LMX9838 (it's a Bluetooth module) uses that communication protocol as a slave device.
And you can implement it with 2 extra GPIOs in any microcontroller.

Ultimately you can implement any type of handshaking with any pair of Master-Slave devices, as long as you have control over a couple of extra GPIOs in both (i.e.: Using uC, FPGA, CPLD...).