When you talk about informing other readers, consider the compiler itself as a reader. If a variable is declared static, that can affect the degree to which optimizations kick in.
Redefining a static variable as extern is impossible, but the compiler will (as usual) give you enough rope to hang yourself.
If I write static int foo; in one file and int foo; in another, they are considered different variables, despite having the same name and type – the compiler will not complain but you will probably get very confused later trying to read and/or debug the code. (If I write extern int foo; in the second case, that will fail to link unless I declare a non-static int foo; somewhere else.)
Global variables rarely appear in header files, but when they do they should be declared extern. If not, depending on your compiler, you risk that every source file which includes that header will declare its own copy of the variable: at best this will cause a link failure (multiply-defined symbol) and at worst several confusing cases of overshadowing.