I believe that:
- the 74HC595 shift register sources current, and
- the TPIC6B595 shift register sinks current.
So: if I want to multiplex some common-anode LED displays, I connect:
- each of the cathode pins of the display units to the corresponding cathode pins of the other display units;
- the current source to the common anode pins of the displays; and
- the current sink to resistors which are in turn connected to the (previously connected-together) cathode pins of the displays.