When making a ‘heicha shemoneh esrei’ where the chazzan will daven until Ha’el hakadosh before the rest of the minyan - is 10 people sufficient or does one need 11 to have 10 people starting shemoneh esrei at the same time for tefila betzibur?
It appears based on the Rama O"C 232:1 that the correct practice could be that all the other people start quietly along with the Chazzan. He does not explicitly answer your question, but everyone starting at the same time would clearly solve your Tefilla BeTzibur problem. However, this does not seem to be the common practice, and the Rama himself says things in other places that complicate this answer.
The Rama O"C 124:2 clarifies that the out-loud recitation need only be through Ha'Kel Ha'Kadosh, but adds that "it would be good for at least one person to be answering Amen" - still implying that (most of) the congregation should be starting along with the Chazzan. For your question, this could go two ways:
- "it would be good for there to be at least one person saying Amen" - but only if there is that extra person over the minimum 10.
- "it would be good for there to be at least one person saying Amen" - even though that means only 9 people would be starting together.
The Kaf HaChaim O"C 124:10 brings two opinions about the above Rama:
- That the above is only true in cases in which there would not be time to do otherwise. If there would be enough time, he says that the Tzibbur should wait until after Kedusha to start their own Tefilla.
- That the above is true all the time, even if there would be time for them to start after Kedusha.
After quoting those opinions, the Kaf Hachaim himself seems to prefer the second opinion because he points out that the Rama seems to prefer it too.
However, the Kaf HaChaim elaborates further that the argument for the Tzibur starting along with the Chazzan is based on the Arizal who says that the "repetition" by definition cannot be started without having had a silent Tefilla. According to some (including the Pri Megadim, based on the Levush), he says, this could really mean that doing this procedure means (according to the Arizal) that the group is not actually mekayem Tefilla BeTzibur, which would be true no matter how many people there are (even 11+).
As a practical answer to your question given our common practice of having the Tzibur start after Kedusha, the Mishnah Berurah 124:10 (and some of the sources above) explicitly refers to 9 people other than the Chazzan, which shows that we are not makpid on having 11 so that 10 can start together.
See OU Torah's article by Rabbi Ari Enkin on the subject of Heicha Kedusha for more sources.