Starship Asterisk*

The VLT or Cerro Paranal Observatory operates 4 large optical telescopes through an Interometer which acts like one large telescope and produces images far better than any individual telescope could.

My thought and question is, Could a similar Interferometer program be run on all telescopes around the world simultaneously to create a single operating structure the relative size of the earth?

In a word, no.

The VLTs, plus ATs, can act as an interferometer (or interferometers) only when instruments such as AMBER and MIDI are used to combine the beams from each telescope. Interferometers at other leading observatories (such as CHARA and at the Kecks) require similarly sublime instruments.

It is, of course, possible to do (optical waveband) interferometry using telescopes as far apart as several thousand km ... provided correspondingly amazing instruments were available. There have been several proposals for space-based interferometers (though not with telescopes even a km apart, much less 1000 km!), but AFAIK none are currently close to being launched or even funded (beyond very early concept stage investigations).

OTOH, proposals for the various next-generation Earth-based optical telescopes - such as TMT, OWL, ELT, etc - all include at least mention of interferometric capability.

What about fiber optics? Not at such dimensions, but let say few km.

Here is a website on the VLTI (VLT interferometer), and here is one on CHARA (the second link in the OP - to a wiki article - contains links to both these, and much more besides).

Note that there are lots of parts to these interferometers; note that most parts are custom-built ... even the telescopes which collect the light that is subsequently combined are special-purpose, designed with interferometry in mind, or retro-fitted (and none of them are cheap).

For CHARA or the VLTI to be extended beyond ~200-300m to, say, a few km, two of the major new things to build/extend would be the 'beam pipes' and the 'delay lines'. As this kind of interferometry depends upon phase, I doubt that fibres could be used to replace 'free space' as the beam pipes. Of course, several extremely good ~km long beam pipes have already been built, and they work very well - the two LIGOs each contain two 4-km beam pipes, for example; so the expertise to build them certainly exists.

There's a different kind of optical interferometry (intensity interferometry) that has far less stringent technical requirements. It doesn't, however, produce 'images', even in a very loose sense of the word. I read a paper on a possible revival of this technique (a special purpose intensity interferometer was built in the 1960s, in Australia) by low-cost equipment add-ons to (TeV gamma) CATs, such as MAGIC and H.E.S.S.; I'll dig up the reference if anyone is interested.

One final note: "images" produced by Earth-based optical interferometry, no matter what the baseline, are limited to ~2" in (angular size). That's about the same as what many amateur astronomers would call good seeing! Oh, and so far, only some stars and AGNs are bright enough to be 'imaged' by this technique anyway.