The ordinals accidentally written on the initial segment of length $\omega$ at the output tape of Ordinal Turing Machines This question implies that (i) all Ordinal Turing Machines start the computation with empty input and no parameters; (ii) we have fixed a particular way to encode an ordinal by an infinite binary sequence (a real).
According to part (iii) of Definition 3.10 in the paper “Recognizable sets and Woodin cardinals: Computation beyond the constructible universe”, a real $x$ is accidentally writable if there is an OTM-program $P$ with empty input such that $x$ is written on the initial segment of length $\gamma$ of the tape [does “the tape” mean the output tape?] at some time. But I want to ask about reals accidentally written on the initial segment of length $\omega$ (the smallest limit ordinal) of the output tape at some time $\tau$, assuming that $\tau$ is not necessarily countable. I will call such reals (or ordinals encoded by reals) “accidentally $\omega$-writable”.
I have read that the accidentally writable reals are exactly the reals in $L$ (part 3 of Lemma 3.13 in the linked paper). Does this imply that the supremum of accidentally $\omega$-writable ordinals is equal to $\omega_1$?
If the answer is “yes”, I need an explanation of why this is possible (that is, how an OTM writes an encoding of an arbitrarily large countable ordinal on the initial $\omega$-segment of the output tape starting from empty input with no parameters). If the answer is “no”, how large is the supremum of accidentally $\omega$-writable ordinals?
 A: If you read the sentence "just before" definition-3.10 then you will see the expression $x \subseteq \gamma$. So if $\gamma$ is some ordinal then it seems to me that the author(s) are talking about some subset $x$ of the $\gamma$. That is a set of ordinals whose where each individual ordinal must be less than $\gamma$. So to define "accidentally writeable reals" specifically, you just need to set $\gamma=\omega$ (in definition-3.10 for the paper you linked).
Further regarding this point:

But I want to ask about reals accidentally written on the initial segment of length $\omega$ (the smallest limit ordinal) of the output tape at some time $\tau$, assuming that $\tau$ is not necessarily countable. I will call such reals (or ordinals encoded by reals) “accidentally $\omega$-writable”.

It seems that no new accidentally writeable real can be generated at some time $\geq \omega_1$. That's because if it was possible, then given some specific real number as input to OTM, it would be possible to halt at some time $\geq \omega_1$, which is impossible [ for this, start enumerating all accidentally writeable reals and when a new accidentally written real matches the "input real" then halt ]. For this, I don't know the reason, but I think that the answer to one of your older questions is relevant in this regard https://mathoverflow.net/questions/372051. I am assuming that the answer given is correct (since I don't understand it).
Edit: Sorry I misremembered the reference I was talking about.
Regarding this point:

I have read that the accidentally writable reals are exactly the reals in $L$ (part 3 of Lemma 3.13 in the linked paper). Does this imply that the supremum of accidentally $\omega$-writable ordinals is equal to $ω_1$?

This should be equal to $\omega^L_1$. It is consistent with set theory that $\omega^L_1=\omega_1$. It is also consistent that $\omega^L_1 \neq \omega_1$. I don't know the reason, but the answer in this question seems to mention this: Existence of bijections in $L$.
