= Solution
The relevant notion is a <root-of-unity Euler system>, which contains auxiliary-prime data as well as the p-power tower. Fix a finite set $S$ of primes containing two but not $p$, and let $W_S$ consist of all <roots of unity> whose orders are prime to every member of $S$. It is a map $\Phi:W_S\to\overline{\mathbb Q}^{\times}$ satisfying these three axioms:
* Evenness and Galois equivariance: $\Phi(\zeta^{-1})=\Phi(\zeta)$ and $\Phi(\sigma\zeta)=\sigma\Phi(\zeta)$ for every $\sigma\in\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$.
* Distribution: for every prime $q\notin S$, $\prod_{\rho\in\mu_q}\Phi(\rho\zeta)=\Phi(\zeta^q)$.
* Auxiliary-prime congruence: if $q\notin S$ and the order of $\zeta$ is prime to $q$, then $\Phi(\rho\zeta)\equiv\Phi(\zeta)\pmod{\mathfrak q}$ for every $\rho\in\mu_q$ and every prime $\mathfrak q$ above $q$; the values are integral at the primes used in the congruence.
The first axiom puts $\Phi(\zeta)$ in the maximal real subfield of $\mathbb Q(\zeta)$. For $\zeta=\zeta_n$, the second axiom is exactly norm compatibility from $K_n^+$ to $K_{n-1}^+$ for $n\ge1$. Auxiliary roots with prime orders different from $p$ are essential: mere norm compatibility of a sequence in the tower is not a verification of all three axioms.
The real <cyclotomic unit> group at a p-power conductor is generated by $-1$ and the <symmetric cyclotomic units> $c_n(a,b)$ with $p\nmid ab$. Galois conjugates are included, since a conjugate of $c_n(a,b)$ is $c_n(ua,ub)$ for an integer $u$ prime to $p$. Thus a prescribed <unit> can be expressed as
$$
\varepsilon=\eta\prod_{j=1}^r d_{a_j}(\zeta_n)^{e_j},\qquad
\eta\in\{\pm1\},\quad p\nmid a_j,\quad e_j\in\mathbb Z,\quad\sum_je_j=0,
$$
where $d_a(z)=z^{-a/2}-z^{a/2}$, with half exponents interpreted in the odd-order root group. Choose $S$ to contain two and every prime divisor of the $a_j$. For $z\ne1$ in $W_S$ put
$$
\Phi(z)=\eta\prod_jd_{a_j}(z)^{e_j},\qquad
\Phi(1)=\eta\prod_ja_j^{e_j}.
$$
Each factor at $z\ne1$ is nonzero because raising $z$ to $a_j$ preserves its order. At $z=1$ each $d_{a_j}$ has a simple zero with first coefficient $-a_j$; the total exponent zero cancels the zeros and signs, giving the stated removable value. Consequently $\Phi(\zeta_n)=\varepsilon$.
We verify the axioms individually. First, $d_a(z^{-1})=-d_a(z)$; the product's sign is $(-1)^{\sum e_j}=1$, so $\Phi$ is even. The unique half-root operation commutes with every <field> automorphism, all exponents are integers and $\eta$ is rational. Hence $\Phi(\sigma z)=\sigma\Phi(z)$, including at $z=1$.
Second, $q\notin S$ is odd and does not divide any $a_j$. For a variable $x$, choose half powers consistently and use
$$
\prod_{\rho\in\mu_q}d_a(\rho x)
=x^{-qa/2}\prod_{\rho\in\mu_q}(1-\rho^a x^a)
=x^{-qa/2}(1-x^{aq})=d_a(x^q).
$$
Here $\rho\mapsto\rho^a$ permutes $\mu_q$, and the product of its half powers is one. Raising these identities to $e_j$ and multiplying gives the distribution identity, since $\eta^q=\eta$. If $x=1$ or $x^q=1$, take the removable values after cancellation; the rational-function identity remains valid there. Thus the distribution axiom holds also at the special roots where the uncancelled factors vanish.
Third, fix $q\notin S$, a root $z$ of order prime to $q$, and a prime $\mathfrak q$ above $q$. Every $\rho\in\mu_q$ reduces to one modulo $\mathfrak q$. If $z\ne1$, its reduction retains its exact order: prime-to-q roots have distinct reductions because their order polynomial is separable in <characteristic> $q$. Therefore $1-z^{a_j}$ is a local <unit>, and all the factors and inverse factors are regular at that reduction. Reduction of the expression for $\Phi(\rho z)$ then gives $\Phi(z)$, since the half roots reduce compatibly. This proves the congruence and local integrality for nontrivial $z$.
At $z=1$, write $\rho=1+T$. The cancelled expression
$$
\eta(1+T)^{-\frac12\sum_ja_je_j}
\prod_j\left(\frac{1-(1+T)^{a_j}}T\right)^{e_j}
$$
is a <unit> <formal power series> over $\mathbb Z_q$, because every $a_j$ and two are q-adic <units>. Its constant term is $\eta\prod_ja_j^{e_j}$. Evaluating at $T=\rho-1$ proves the remaining congruence modulo every prime over $q$. This also covers $\rho=1$. Hence \b[every real <cyclotomic unit> at any layer is a value of an <Cyclotomic Euler system> satisfying all three axioms].
For completeness, at a nontrivial root $z$ every ratio $d_a(z)/d_b(z)$ is a global <unit>. After removing a root-of-unity factor, both it and its inverse are geometric sums because $a,b$ are invertible modulo the order of $z$. Thus the values used at nontrivial roots really are <units>, not just nonzero algebraic numbers. Values at one may be rational S-units, which is compatible with the definition.
The auxiliary norm factor can also be read off explicitly. If $z$ has order $m>2$ and $q\nmid m$, then for $\rho\ne1$ of order $q$,
$$
N_{\mathbb Q(\mu_{mq})^+/\mathbb Q(\mu_m)^+}\Phi(\rho z)
=\prod_{\rho'\in\mu_q\setminus\{1\}}\Phi(\rho'z)
=\frac{\Phi(z^q)}{\Phi(z)}=\Phi(z)^{\operatorname{Fr}_q-1}.
$$
Here the <arithmetic Frobenius> sends $z$ to $z^q$. This convention makes the sign of the Euler factor explicit, while the case of a repeated prime $p$ gives the exact tower norm relation already proved in Question 2.
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