ADDITIONAL SYSTEM INFORMATION :
Windows 10 Pro

A DESCRIPTION OF THE PROBLEM :
When first compiling two interdependent annotation classes and an annotation processor and then in the second step a class bearing one of the annotations the processor is focusing on, using same processor for annotation processing during javac compilation, the following exception appears:

...
An annotation processor threw an uncaught exception.
Consult the following stack trace for details.
java.lang.annotation.AnnotationTypeMismatchException: Incorrectly typed data found for annotation element public abstract org.example.MySecondAnnotation org.example.MyFirstAnnotation.secondAnnotation() (Found data of type org.example.MySecondAnnotation)
...

This works on JDK 8 and to some extent in 11 (with default target 11, not with '--release 8') and not at all in 17 and 21.

REGRESSION : Last worked in version 8

STEPS TO FOLLOW TO REPRODUCE THE PROBLEM :
On JDKs 17, 21 (same works on 8, 11):
$ javac org/example/MyFirstAnnotation.java org/example/MySecondAnnotation.java org/example/Processor.java 
$ javac -processor org.example.Processor -cp . org/example/AnnotatedClass.java

On JDK 11:
$ javac --release 8 org/example/MyFirstAnnotation.java org/example/MySecondAnnotation.java org/example/Processor.java 
$ javac --release 8 -processor org.example.Processor -cp . org/example/AnnotatedClass.java


EXPECTED VERSUS ACTUAL BEHAVIOR :
EXPECTED -
Compilation works, second command prints:

Processor ran!
[org.example.MyFirstAnnotation]
[errorRaised=false, rootElements=[org.example.AnnotatedClass], processingOver=false]
@org.example.MyFirstAnnotation(secondAnnotation=@org.example.MySecondAnnotation(stringArray={""})) -> @org.example.MySecondAnnotation(stringArray={""})
warning: No SupportedSourceVersion annotation found on org.example.Processor, returning RELEASE_6.
warning: Supported source version 'RELEASE_6' from annotation processor 'org.example.Processor' less than -source '11'
Processor ran!
[]
[errorRaised=false, rootElements=[], processingOver=true]
2 warnings

ACTUAL -
Processor ran!
[org.example.MyFirstAnnotation]
[errorRaised=false, rootElements=[org.example.AnnotatedClass], processingOver=false]
warning: No SupportedSourceVersion annotation found on org.example.Processor, returning RELEASE_6.
warning: Supported source version 'RELEASE_6' from annotation processor 'org.example.Processor' less than -source '17'
2 warnings


An annotation processor threw an uncaught exception.
Consult the following stack trace for details.
java.lang.annotation.AnnotationTypeMismatchException: Incorrectly typed data found for annotation element public abstract org.example.MySecondAnnotation org.example.MyFirstAnnotation.secondAnnotation() (Found data of type org.example.MySecondAnnotation)
        at jdk.compiler/com.sun.tools.javac.model.AnnotationProxyMaker$ValueVisitor$1AnnotationTypeMismatchExceptionProxy.generateException(AnnotationProxyMaker.java:271)
        at java.base/sun.reflect.annotation.AnnotationInvocationHandler.invoke(AnnotationInvocationHandler.java:89)
        at jdk.proxy3/jdk.proxy3.$Proxy3.secondAnnotation(Unknown Source)
        at org.example.Processor.process(Processor.java:23)
        at jdk.compiler/com.sun.tools.javac.processing.JavacProcessingEnvironment.callProcessor(JavacProcessingEnvironment.java:1023)
        at jdk.compiler/com.sun.tools.javac.processing.JavacProcessingEnvironment.discoverAndRunProcs(JavacProcessingEnvironment.java:939)
        at jdk.compiler/com.sun.tools.javac.processing.JavacProcessingEnvironment$Round.run(JavacProcessingEnvironment.java:1267)
        at jdk.compiler/com.sun.tools.javac.processing.JavacProcessingEnvironment.doProcessing(JavacProcessingEnvironment.java:1382)
        at jdk.compiler/com.sun.tools.javac.main.JavaCompiler.processAnnotations(JavaCompiler.java:1234)
        at jdk.compiler/com.sun.tools.javac.main.JavaCompiler.compile(JavaCompiler.java:916)
        at jdk.compiler/com.sun.tools.javac.main.Main.compile(Main.java:317)
        at jdk.compiler/com.sun.tools.javac.main.Main.compile(Main.java:176)
        at jdk.compiler/com.sun.tools.javac.Main.compile(Main.java:64)
        at jdk.compiler/com.sun.tools.javac.Main.main(Main.java:50)


---------- BEGIN SOURCE ----------
package org.example;

public @interface MyFirstAnnotation {
    MySecondAnnotation secondAnnotation() default @MySecondAnnotation;//(stringArray = "");
}

------------------------------------------------------------------------

package org.example;

public @interface MySecondAnnotation {
    String[] stringArray() default "";
}

------------------------------------------------------------------------

package org.example;

import java.util.Set;

import javax.annotation.processing.AbstractProcessor;
import javax.annotation.processing.RoundEnvironment;
import javax.annotation.processing.SupportedAnnotationTypes;
import javax.lang.model.element.Element;
import javax.lang.model.element.TypeElement;

@SupportedAnnotationTypes("org.example.MyFirstAnnotation")
public class Processor extends AbstractProcessor {

    public boolean process(final Set<? extends TypeElement> annotations,
                           final RoundEnvironment roundEnv) {

        System.out.println("Processor ran!");
        System.out.println(annotations);
        System.out.println(roundEnv);

        for (Element element : roundEnv.getElementsAnnotatedWith(MyFirstAnnotation.class)) {
            MyFirstAnnotation firstAnnotation = element.getAnnotation(MyFirstAnnotation.class);
            MySecondAnnotation secondAnnotation = firstAnnotation.secondAnnotation();
            System.out.println(firstAnnotation + " -> " + secondAnnotation);
        }
        return true;
    }
}

------------------------------------------------------------------------

package org.example;

/*
 * Uncomment one of these lines and the annotation processor starts working.
 */

//import org.example.MySecondAnnotation;
@MyFirstAnnotation(
        //secondAnnotation = @MySecondAnnotation
)
public class AnnotatedClass {
}

---------- END SOURCE ----------

CUSTOMER SUBMITTED WORKAROUND :
Either of the following works:

1.) In AnnotatedClass, change @MyFirstAnnotation to @MyFirstAnnotation(secondAnnotation = @MySecondAnnotation)

2.) In AnnotatedClass, redundantly import org.example.MySecondAnnotation

3.) In MyFirstAnnotation, change the default from @MySecondAnnotation to @MySecondAnnotation(stringArray = "")

FREQUENCY : always