White paste/chunks in coolant system

My car guy and I are completely stumped.

Car Details: 2004, Scion xa, 70K, standard, well-kept

History: Heat stopped working, upon inspection of radiator cap found a white thick pasty substance. Found leaks in radiator. Car did not overheat at anytime.

Work: Replaced radiator, flushed system (chucks/pieces of white substance came out during flush), added new coolant, heat still did not work. Drained and flushed system again, replaced with straight water, ran car and it heated well, drained. Filled again with radiator flush and water, ran and heated well. Left water and flush in system 2 days. Less than one hour of driving resulted in complete radiator failure (we actually believe we had a defective radiator). Replaced radiator. Flushed (more white chunks/pieces came out) and added new coolant, heat is not working again.

It appears that whatever this substance is that it solidifies in coolant but dissolves in water (heat works fine running system on straight water). It does not seem to be building up under the radiator cap any longer, but does appear to be building up in the heater core or at least that’s my theory. My mechanic has never seen anything like this before and he has also asked his circle of mechanic friends who have never experienced this.

We do not believe this is a blown head gasket. I’d welcome any thoughts. Thanks.

So what was it? Did it go away after more flushes?