ULN 2003 open collector
Figure 1. Internal diagram of the ULN2003. Source: Wikipedia ULN2003A.

Note the internal snubber diodes. This makes your circuit look like this:







simulate this circuit &ndash; Schematic created using CircuitLab

Figure 2. The actual schematic showing the internals of the ULN2003.

SW1 represents your power supply. When up, as shown, the snubbers are connected to the switched +12 V. When SW1 is reversed, representing the PSU being switched off, a path to ground exists through all the components in the rest of the circuit. It's obviously low enough to allow your relay to pick up.

The solution:







simulate this circuit

Figure 3. Connect the snubber COM to the always-on +12 V.

This will eliminate the sneak path to ground. More importantly, it will return the relay current to the correct power source - the "always-on" +12 V.