I’m a big fan of 7812 and associated regulators. They are NASA reliable, cheap, and usually stupid simple to wire up and use. Except! There have been counterfeit ones sneaking into the market.
All linear regulators require an input voltage higher than their output voltage. A datasheet somewhere might claim 1.5 volts, I have found that’s not enough. More like 3 (or more) volts if you never want to experience dropout.
Sounds to me like you would want an ordinary 7812 and just insert a diode into the ground leg so that your 12 volts output becomes 12.7 volts. In other words, the ground of the 7812 references .7 volts instead of zero volts. Or two series diodes if you want a 13.4 volt output.
Yes, the 7812 was the 1st thing, well, 2nd thing I thought of, as I’ve used the 7815 and 7915 (and 7818 & 7918) in split power supplies for audio op amps, many moons ago.
However, the dropout behavior is my question: If using a 7812 and the input to it drops to, say, 12.7 volts, or even 12.0v, does the output turn off? Drop to who knows what lower voltage, possibly very unstable?
I’m thinking a zener-transistor circuit might perform better in this regard. (?)