Legal and regulatory environment

The basic principle in taxation is that the income is always taxed at the situs of taxation.  This means that income is usually taxed at the source of the income.  Every time income, as in this case, is earned in a France and in Austria then the taxing authority of that area has the right to collect tax on that realized income.  It must be important to remember that the right to tax the income realized is only for the actual realized income.  It is fundamental in taxation that income that has not been realized such as stock dividends or bonds without yield do not fall under income for purposes of taxation.
Another important principle is that foreigners are also liable for taxes under the local taxing authority under where they earn their income.  It does not basically follow that just because a person, in this case Frida, is a foreign national that she owes taxes to her country of citizenship.  While she does owe her country of citizenship a certain percentage of taxes, she is only liable for the income that was earned exclusively within the taxing jurisdiction of that country save in cases where she remits the income back to her country and such is subject to another tax but not based on income tax.
In this scenario, Frida is liable to pay taxes to her country, Chile, to her place of business, the United States, to the country where her chateau is located, France and if her bonds earn any returns, Australia.  The reason for this, as explained earlier, is that Frida is liable to the countries where she earned her income.  Since she is a citizen of Chile, she owes Chile any taxes on any income that she has earned in Chile.  She is liable to the United States for any income that she may have earned there from her realized income from her Savings and Drug Corporation.  Her liability in France is based on her income from the chateau that she owns there.  Finally, in the event that the bonds which she owns in Australia offer any return or interest yield, she has to pay the proportional amount of taxes to the local taxing authority there.

The countries will only be able to tax Frida on the basis of the actual income that was realized within their jurisdiction.  It is a fundamental principle of taxation that only income actually realized can be the basis of taxation.  In this case, France can tax the rental income generated from the chateau because it constitutes realized income.  Any other income that is generated by Savings and Drug Corporation is taxed in the local taxing authority where the income is generated.  It must be remembered also that Frida is not necessarily liable to any of these countries wherein the Savings and Drug Corporation earns income because the corporation is a separate juridical entity that is taxed separately from the individual.  She is only liable for the income which she personally received from the company.  This is usually at the head office of the corporation.
As previously mentioned, it is a principle in international private law that the local taxing authority has the right to tax any income that is generated within its jurisdiction.  By doing any sort of commercial or business transaction in a foreign jurisdiction that earns income, an entity is supposed to pay the proportional amount of taxes to the local authority.  This is under the principle that a benefit was gained from the business in that country to which the local taxing authority has a right to exact taxes from.
Finally, in the case of double taxation treaties, the taxes that have already been collected on income generated abroad is generally not taxed again by the home country.  In case the double taxation treaty provides that income once taxed in a foreign jurisdiction is no longer subject to local taxes then the income that Frida earned in the United States and France can no longer be taxed in Chile or in certain cases be reduced according to a certain table.
References:
Hoffman, Phillipe and Kathryn Norberg (1994), Fiscal Crises, Liberty, and Representative Government, 1450-1789, p. 238.
Zelizer, Julian E. Taxing America: Wilbur D. Mills, Congress, and the State, 1945–1975. Cambridge, U.K.: Cambridge University Press, 1998.
Dick Netzer, Economics of the Property Tax (1966); J. F. Due, Government Finance (4th ed. 1968); C. S. Shoup, Public Finance (1969); H. M. Groves, Financing Government (7th ed. 1973); C. Webber and A. Wildavsky, A History of Taxation and Expenditure in the Western World (1987).