I must admit, that I have become obsessed with this mysterious song-snippet too. Since many years I have been following this thread, but I did not know how to contribute and I was hoping that someone will solve it (in the end).
Since no one solved it by now, I want to tell you my (lengthy) thoughts on this. Think of it as some kind of summary, omitting the ruled out songs and hopefully concentrating on the interesting bits and facts for further searching.
I think "deevo0" made some good remarks on how to structure further musical investigations. By now, it should be clear that this song is not an easy task to solve. Perhaps it was never officially released or the record company was very very small or the company folded up during the production or release. Somehow this recording vanished into obscurity and that is basically all we know.
Another thing is, I'm not really sure whether it is a good production or a low-budget one. Since it is an old tape recording from a radio station, I would like to know the recording equipment (big stereo system vs. small portable stereo). Perhaps the owner of this recording could send us snippets of the other recordings on that tape. Especially the songs which were recorded directly before and after the mysterious song. This could shed some light upon overall sound quality and it could put "stay (second time around)" into a larger perspective. For example, if the surrounding songs are rarities in itself or if these songs are from german artists or german record companies.
Furthermore I have read the information given by the original tape owner that the song was either recorded in 1984 (Hessen 3, HR3) or 1986 (NDR 1). How can he be so sure? But if he is right, he should remember the name of the radio shows. Obviously it must have been a show for a special interest crowd, preferably aired late in the evening.
In an earlier posting (around 2006) it says, someone suggested that the song was recorded off the radio show "Songs vom Synthesizer" hosted by Rainer Sauer. Is it true or merely speculation?
Rainer Sauer was even contacted back in 2006 and he answered that "Stay" was recorded by "K. Barré" but it was not the song obviously.
Interestingly enough Rainer Sauer aired the old series again in 2009. Under the name 'Synthesizer Studio' some (or all?) old radio shows "Sounds vom Synthesizer" were available as an online stream from Webradio ZONO. Sadly, I missed the opportunity to listen to these shows. Here are some Links to the show and Rainer Sauer in german language:
http://www.soundsvomsynthesizer.de/
http://a-u-t-o-b-a-h-n.blogspot.de/
https://twitter.com/RainerSauerJena
Regarding the discussions on the show "Sounds vom Synthesizer" it must have had a very nerdish approach on synthesizer music and electronic sounds. But, as Rainer Sauer stated in his posting (email) from 2006, the show often played music from ZYX (german record company with mainly italo-disco style artists) and from the new artist section of the show emerged the band Camouflage (The Great Commandment). Taking these facts into consideration, it might still be possible that "Stay" was played on "Sounds vom Synthesizer". The show existed from 1983 until late 1985.
Having said that, I must admit, that I don't think "Stay" was recorded or even played in 1984. Many early synth-pop or dark wave (electro wave) artists used analog synths and drum machines. I looked through many old vinyl records from that time, mostly rare 7-inch singles from flea markets and car boot sales. Many early recordings have very fat sounding synth basslines and heavy artificial sounding drumcomputer use (Roland TR-808) throughout the tracks. Our mysterious song "stay" has no audible bassline at all and the drumcomputer sounds fairly natural. All in all it sounds very "polished". Furthermore the lush sounding synth-glitter at the end of the song is very characteristic. It sounds digital and is probably a Yamaha DX 7 (introduced in 1983) with its "Tub Bell" (Tubular Bell) preset. After its release in 1983 the new digital sounding DX 7 made a huge impact on the synth-scene and somehow gradually superseded the analogue sounding synths. On top of this, during the mid-eighties synth-pop started to decline. If "Stay" would have been recorded as early as 1984, I think every A&R division of a record label would have known that it might have been a minor hit-single with eventual chart success at the time.
To sum it up - "Stay" has:
No Funky Bass
No or very quiet Guitar
No Trumpets or Saxophone
No Handclaps (disco -> Italo)
No Orchestral Stab (disco- Italo)
No fat Disco Beat
So, if "Stay" is probably from 1986 as the original owner suggested, which NDR1 Radio Show did he record by the time?
As we know, it was certainly played by a german radio station! This raises another question for me - why should a german station play the song? How would they know about this record? If the band was german this question would be easy to answer, but the singer does not sound very german to me. (I am german, by the way). During the 1980s a german singer was easy to identify. (For example:
http://www.youtube.com/watch?v=vkLdf9xzcc4) And even the widely known Alphaville or Camouflage don't sound exactly as if their vocalists were English native speakers! Am I right?
Unfortunately, I don't know very much about 1980s record company structures or distribution politics. Therefore I'm speculating (once again)! If the record company was german or a big global player with global distribution it would have been possible that a german radio station played the song due to promotion. But if it was a small label from the UK, the record would not have been played, promoted or distributed in Germany if the record didn't have some impact on the charts in its home country. This makes me think, perhaps it was a german band with an English singer. Or the producer of the song was german with good relations to a german record company and/or radio stations (NDR 1). To put it simply, how do we connect this recording to a german radio station? Why does or did it remain completely unknown to the rest of the world?
When I heard the song for the first time I thought this one would be easy to solve. The singers voice sounded familiar and a million references and possibilities popped up in my head. Eventhough "Stay" is very different from the following two songs, my first "musical reference" thoughts were from the years 1986/1987. Probably some synth sound similiarities made me think of:
http://www.youtube.com/watch?v=1i9lBLj7GnY
(Mike & the Mechanics - Silent Running)
http://www.youtube.com/watch?v=G6BFJJPF1Xk
(Humpe & Humpe - Careless Love)
I know these Links aren't very helpful, but after listening to the usual suspects I ruled out for myself the following bands because their sound and vocalists sound different to me: Ultravox, Midge Ure, Japan, David Sylvian, China Crisis, Blancmange, Naked Eyes, Classix Nouveaux, Heaven 17, Pseudo Echo, John Foxx, Icehouse, Lotus Eaters, Fiat Lux, Al Corley
And I ruled out for myself many suspects from my vinyl obscurities because of sound and voice:
A Popular History Of Signs, Voice of America, Zerra One, Vitamin Z, H2O, Boxbury Beat, Opposition, Bourgie Bourgie, Birch and Hallam, Music For Pleasure, Fra Lippo Lippi, The Expression, Models
Concerning the voice of the singer it comes close to the following bands:
Car Crash Set, Furniture, The Light (Box Of Toys)
That's all! [:I]