Interstellar construction between star systems

Basic scenario is that you have two stars surrounded by Dyson spheres, assuming time and resources is no object would it be possible to build a bridge between two the two sphere's and effectively anchor them relative to each other?

Or would their slightly different orbits/movements in the galaxy make it impossible, or the shear length of the connection cause it to collapse/bend under its own weight.

In addition what would happen to the galaxy if you applied this process to every star in it? Apart from the fact you would need more materials that actually exist would it simply turn the galaxy into a blackhole.

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Interstellar construction between star systems

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