Wait for [NSAlert beginSheetModalForWindow:...];

Here is a NSAlert category that solves the issue (as suggested by Philipp with the solution proposed by Frederick and improved by Laurent P.: I use a code block instead of a delegate, so it is simplified once again).

@implementation NSAlert (Cat)

-(NSInteger) runModalSheetForWindow:(NSWindow *)aWindow
{
    [self beginSheetModalForWindow:aWindow completionHandler:^(NSModalResponse returnCode)
        { [NSApp stopModalWithCode:returnCode]; } ];
    NSInteger modalCode = [NSApp runModalForWindow:[self window]];
    return modalCode;
}

-(NSInteger) runModalSheet {
    return [self runModalSheetForWindow:[NSApp mainWindow]];
}

@end

We created a category on NSAlert to run alerts synchronously, just like application-modal dialogs:

NSInteger result;

// Run the alert as a sheet on the main window
result = [alert runModalSheet];

// Run the alert as a sheet on some other window
result = [alert runModalSheetForWindow:window];

The code is available via GitHub, and the current version posted below for completeness.

---

Header file NSAlert+SynchronousSheet.h:

#import <Cocoa/Cocoa.h>


@interface NSAlert (SynchronousSheet)

-(NSInteger) runModalSheetForWindow:(NSWindow *)aWindow;
-(NSInteger) runModalSheet;

@end

---

Implementation file NSAlert+SynchronousSheet.m:

#import "NSAlert+SynchronousSheet.h"


// Private methods -- use prefixes to avoid collisions with Apple's methods
@interface NSAlert ()
-(IBAction) BE_stopSynchronousSheet:(id)sender;   // hide sheet & stop modal
-(void) BE_beginSheetModalForWindow:(NSWindow *)aWindow;
@end


@implementation NSAlert (SynchronousSheet)

-(NSInteger) runModalSheetForWindow:(NSWindow *)aWindow {
    // Set ourselves as the target for button clicks
    for (NSButton *button in [self buttons]) {
        [button setTarget:self];
        [button setAction:@selector(BE_stopSynchronousSheet:)];
    }

    // Bring up the sheet and wait until stopSynchronousSheet is triggered by a button click
    [self performSelectorOnMainThread:@selector(BE_beginSheetModalForWindow:) withObject:aWindow waitUntilDone:YES];
    NSInteger modalCode = [NSApp runModalForWindow:[self window]];

    // This is called only after stopSynchronousSheet is called (that is,
    // one of the buttons is clicked)
    [NSApp performSelectorOnMainThread:@selector(endSheet:) withObject:[self window] waitUntilDone:YES];

    // Remove the sheet from the screen
    [[self window] performSelectorOnMainThread:@selector(orderOut:) withObject:self waitUntilDone:YES];

    return modalCode;
}

-(NSInteger) runModalSheet {
    return [self runModalSheetForWindow:[NSApp mainWindow]];
}


#pragma mark Private methods

-(IBAction) BE_stopSynchronousSheet:(id)sender {
    // See which of the buttons was clicked
    NSUInteger clickedButtonIndex = [[self buttons] indexOfObject:sender];

    // Be consistent with Apple's documentation (see NSAlert's addButtonWithTitle) so that
    // the fourth button is numbered NSAlertThirdButtonReturn + 1, and so on
    NSInteger modalCode = 0;
    if (clickedButtonIndex == NSAlertFirstButtonReturn)
        modalCode = NSAlertFirstButtonReturn;
    else if (clickedButtonIndex == NSAlertSecondButtonReturn)
        modalCode = NSAlertSecondButtonReturn;
    else if (clickedButtonIndex == NSAlertThirdButtonReturn)
        modalCode = NSAlertThirdButtonReturn;
    else
        modalCode = NSAlertThirdButtonReturn + (clickedButtonIndex - 2);

    [NSApp stopModalWithCode:modalCode];
}

-(void) BE_beginSheetModalForWindow:(NSWindow *)aWindow {
    [self beginSheetModalForWindow:aWindow modalDelegate:nil didEndSelector:nil contextInfo:nil];
}

@end

The solution is to call

[NSApp runModalForWindow:alert];

after beginSheetModalForWindow. Also, you need to implement a delegate that catches the "dialog has closed" action, and calls [NSApp stopModal] in response.