AK,
All op amps, regulators, references, InAmps, etc power the gain stages with current sources and sinks
in order to keep the performance relatively constant with different power supply voltages. Note the AD8657
is spec'ed to operate from 2.7V to 18V. To bias the sources and sinks requires a sloppy voltage reference,
similar to a band gap ref. But refs usually need a startup circuit. They might not work with a rise time
that is too fast or too slow. They might not always work at cold or at hot. Sounds like the startup circuit
is rise time dependent, not a latchup problem. It might also be a loading problem or circuit problem,
your schematic is too hard to read. Can you attach a PDF?
Harry