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And with a heavy load the windings will only source current into the load, not into each other.I would have said that if you have two identical transformers in parallel, with the primaries in parallel as well, that would usually be OK. There might be a problem with large transformers as they have better regulation, which means that their output drops less at full load.
If you are rectifying the output of the transformers, you can always put a rectifier on each transformer and parallel the outputs of the rectifiers. There will be no problem doing that.
It is certainly common practice to parallel two secondary windings on one transformer.
Any unbalance in voltage will cause an unbalance in the load distribution between the two transformers, potentially overloading one transformer.And with a heavy load the windings will only source current into the load, not into each other.
If a secondary can source 1A without overheating, I guess it could also sink 1A without overheating.
I finally bit the bullet and solved the two loop equations for this setup, for I1 and I2, and checked it with a spreadsheet.That's not to say you couldn't get by by with paralleling two identical transformers, but it's not good engineering practice.