Obviously, it's not showing the current time yet, but just the number of minutes and seconds since the last DCF77 reception error. It seems that the HT16K33 LED controller IC on the back of the 7-segment display interferes with the DCF77 signal. I don't even have to send commands to the IC, simply powering it is enough to cause the issue. I tried different orientations and putting a metal shield in between the two but it didn't help. The only thing that helps is distance. Once I put the DCF77 antenna and the HT16K33 IC about 20cm apart it works. You can see that it worked for more than 49 minutes before I took the picture and stopped the test.
I planned to put all of the clock parts into a simple case made from three pieces of plastic. Well, now I'll need two cases, because I don't want a clock so big that I can get 20cm distance between the IC and the antenna.