photodiode amplifier
Yes, there is a way to split it.  Some people just use two biggish equal resistors across the supply, and use the middle as ground (it really is a reference), and the negative terminal would be your negative, and the positive, your positive.

There are some disadvantages, not the least of which is the lack of a low-impedance ground that can sink current

A better approach is a chip like the TLE2426. Just read the data sheet.  An alternative is to just use two DC-to-DC converters, one to +6, and one to -6, off your 12 volts.

Also, once you have +/- 6V, you could use the appropriate voltage regulators to bring that down to +/- 5V.  For example, http://www.digikey.com/product-detail/en/MC78L05BP-AP/MC78L05BP-APMSCT-ND/804696 and http://www.digikey.com/product-detail/en/MC79L05BP-AP/MC79L05BP-APMSCT-ND/804701.  I haven't verified the drop outs (i.e., how much voltage you need to feed them to get your required output voltage), or asked about your current requirements, but your search term would be LDO regulator if these don't suit.