= Solution
The real <canonical variables> $x^m,p_m$ describe the center-of-mass position and total <momentum> of the <open string>. The nonzero <string oscillators> are complex <Fourier modes> with $\alpha_{-k}^m=(\alpha_k^m)^*$; the <Lagrange multipliers> obey $\lambda_{-j}=\lambda_j^*$. The <Virasoro constraints> satisfy $L_{-j}=L_j^*$, so the multiplier term is real.
For each $k>0$, the oscillator kinetic term differs from the manifestly real expression
$$
\frac{i}{2k}\bigl(\dot\alpha_k\cdot\alpha_{-k}
-\alpha_k\cdot\dot\alpha_{-k}\bigr)
$$
by $\frac{i}{2k}\frac{d}{dt}(\alpha_k\cdot\alpha_{-k})$. Thus the written <action> is real up to a boundary term, which does not change its <symplectic form> or bulk dynamics. Adding the corresponding endpoint term makes reality exact.
The independent nonzero <Poisson brackets> are
$$
\boxed{\{x^m,p_n\}=\delta^m_n,\qquad
\{\alpha_j^m,\alpha_k^n\}=-ij\,\eta^{mn}\delta_{j+k,0},\quad j,k\neq0.}
$$
All brackets between these independent center-of-mass and oscillator variables vanish. The dependent zero mode is $\alpha_0^m=p^m/\sqrt{\pi T}$, so, if it is used, $\{x^m,\alpha_0^n\}=\eta^{mn}/\sqrt{\pi T}$.
The <Fourier coefficients> of the <Virasoro constraints> are
$$
L_j=\frac12\sum_{k\in\mathbb Z}\alpha_{j-k}\cdot\alpha_k,\qquad
L_0=\frac{p^2}{2\pi T}+\sum_{k=1}^{\infty}\alpha_{-k}\cdot\alpha_k.
$$
Their <Poisson brackets> are
$$
\boxed{\{L_j,L_k\}=-i(j-k)L_{j+k}.}
$$
They form the classical <Witt algebra>, with no <Virasoro central extension>. In particular they are <first-class constraints>, closing on the constraint surface, and generate the remaining <worldsheet diffeomorphisms> rather than independent physical excitations.
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