I think this is Ostropales. I was thinking about Stictis conotremoides Sherwood. Perhaps you can try looking at that? I collected this once in Canada. Do you know about these papers on "optional lichenization" by Mats Wedin? He was writing about Conotrema urceolatum (= Stictis urceolata) which is a lichen, but can develop as a saprobic fungus (no thallus) depending on environmental factors. I wonder if your fungus is not a saprobic state of Stictis urceolata.

Hello Jason,
thank you very much for your contribution and the species proposal(s).
I will try at home to fiind out more (am on the journey back).
I then will try to find Wedins paper.
Best regads, Lothar

here is the Wedin article

Here is a link to the protologue of Stictis conotremoides. Note that I don't think Wedin mentions this species but I suspect this is related to the issue.

https://www.biodiversitylibrary.org/item/340945#page/265/mode/1up

Hello Jason,
sorry for the late reply - but we jaust came back home from our journey yesterday. 
Thank you very much for the Wecin paper and the link.
Unfortunately, the link seems not to work properly (at least in the moment - I will try again later). 
The site is testing that I am not a bot - but as it seems not successfully. It begins again and again to test ...
Best regards, Lothar

Well, that's a pain in the butt. That's happened to me once before but it cleared up. I'd try it again. In the meantime, here's screenshots of the description. I think there might be differences between yours and this description but maybe close? Do you think yours could be a Stictis?

thank you very much Jason!
I will write again when I find time again to go to the subject.
In the moment, there are lots of things to do first.
Yours, Lothar

Hello Jason,

I just took a look at the publications you sent me - thanks again for this!
And yes: Stictis conotremoides (urceolatum) seems to be a very good hint, close to my find.

What makes me hesitate is ecology - and I do not mean the problem of lichenization and saprotrophy.
No - Wedins collections are from Northern Norway, and Sherwoods from Arctic Canada.

Do you know of Stictis urceolata-collections from the Mediterranean climate zone?

I could still re-examine a part of the specimen. What do you think could I still do to contribute to more clarification?

Yours, Lothar

Got it. Yes, Mediterranean... so maybe not Stictis urceolata. Maybe when you reexamine the specimen, confirm that it's Stictis? Do you have Sherwood's various papers dealing with Stictis and Schizoxylon species? The dark blue-gray disc is striking.

After some search I find a specimen from Taiwan, on Miscanthus, that could have a similarly warm climate. It seems to be sequenced. 
https://asianjournalofmycology.org/pdf/AJOM_3_1_3.pdf
In the text the spores fit well, and the macrofotos do this, too. But in the table of fotos there are spores of a normal Stictis (long and thin). A mistake in the paper ...???

I think Stictis urceolata ist a good hint still.

What should be the proof for a Stictis? 
I will try to do a section of the apothecium, and if I get better pcitures, I will show.

I'll look at the books I have at home and get back to you. In the meantime, I think the big distinction for Stictis vs other genera like Schizoxylon is that the hymenium is always separated from the inner surface of the "margin" when the specimen is dry. So, looking at an apothecium through your dissecting scope, there should be a space between the hymenium and the margin. I hope this is clear. Do you have Sherwood's 1977 monograph in Mycotaxon? It's that link to BHL I sent you earlier. I hope that it works for you this time! There's illustrations of what I'm talking about in the Introduction. Definitely try to make a section of an apo.

Yes, the spores in the image in that paper are quite narrow. They don't look anything like 3-4 um wide like they wrote in the description.

thank you, Jason?
I will try and then (perhaps) write again ...
Yours, Lothar

Hello Jason,
this morning I did a section of the last living apothecium I still had in "moist chamber". 
I hope the results bring more clearity - although I must admit I am not the very master of accurate cutting small samples.
I show pictures of the unsquashed cut, of the excipular structures, the periphyses (that do not seperate well from the hymenium even after squashing) and - another picture of the spores. Those seem to be partly also transversely septate (?), I do not know if this would change a lot (?).
Yours, Lothar

P.S. I just looked at your screenshot from Sherwood again - and see what you mean with the space. In my living specimen I did not see this, and also not in the dried one. I made a section of one of the few dried apothecia but was not able to make a foto (crumbling pieces ....).
Sorry ...
On the other hand, I find the similarieties to Stictis urceolata strong.

P.S. You sent me a link to Sherwood 2 - could you also provide me with Sherwood 1?

Hi Lothar,
Here is the link to part 2: https://www.biodiversitylibrary.org/item/340945#page/233/mode/1up
Your images are very helpful! I see that the spores even have longitudinal septa. I think that if you didn't see the hymenium pulling away from the inner surface of the margin then it may not be Stictis. Let's try Schizoxylon. There's treatments in part 1 and 2.

Hello Jason,
thank you. I will try and write again.
First: you again sent me part 2 - what misses is part 1 (even if it may not any more be necessary for this item).
Thanks again and best regards, Lothar

Oops! Sorry.
Here is the link to part 1: https://www.biodiversitylibrary.org/item/340944#page/13/mode/1up
After seeing your new images I think that it makes better sense that this might be a Schizoxylon, but going through 1 and 2 (just doing a quick scan, looking at images) I don't know what species. Perhaps you will have good luck keying it out.

thanks, Jason, for the link - I just downloaded.
Few minutes ago, I took a quick try with the key of Schizoxylon in Part 2 - and was not very lucky, too. 

Okay. Well, there may not be much satisfaction right now. Keep trying. Perhaps there are more recent Schizoxylon names in IF or MycoBank, and one of them was described from Mediterranean material? These are always tricky for me, and I'm often not confident in my determinations. It could also be that you have a new species! haha!