It is possible for the RL78/G13 to control an SPI. Set the SAU of the RL78 to CSI (3-line serial), and also prepare one port for using for the CS signal. Make the CS signal low before starting SPI communication, carry out communication using CSI, and after communication has finished drive the CS signal high.
Also, please note the following if using the RL78/G13 or RL78/G12 as a slave of the SPI. When using as a slave of the SPI, after selection using the CS signal from the master, communication won't actually be possible until the program is finished processing. Thus, a wait is required after selection by the CS signal from the master. This is described in an RL78/G12 application note, so please refer to it.