Is this an ARMA(2, 1) process? I am puzzled by an equation,
$$
y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + u_t + \varepsilon_t - \varepsilon_{t-1},
$$
where $u_t$ and $\varepsilon_t$ are independent white-noise processes.
Is this an ARMA(2, 1) process? Or does the fact that there are two shocks at each timestep (but only one carried over) change this?
Secondly, can $\phi_1$ and $\phi_2$ be estimated consistently using 2SLS? I thought that $y_{t-2}$ would be a valid instrument for $y_{t-1}$, but then $y_{t-2}$ is also a regressor variable.

Your model
$$
(1-\phi_1B-\phi_2B^2)y_t=u_t+(1-B)\epsilon_t \tag{1}
$$
is ARMA(2,1) since the right hand side is clearly an MA(1) process since its autocovariance function cuts off for lags larger than 1. Hence, the right hand side can be represented by
$$
(1-\theta_1 B)v_t \tag{2}
$$
where $v_t$ is another white noise process.
Equating the autocovariance of the right hand side of (1) to that of (2) at lag 0 and 1 yields a set of two non-linear equations
\begin{align}
\sigma_u^2+2 \sigma_\epsilon^2 &= \sigma_v^2(1+\theta_1^2)
\\ -\sigma_\epsilon^2 &= -\theta_1\sigma_v^2
\end{align}
which can be straightforwardly solved for $\sigma_v^2$ and $\theta_1$.  There are two solutions. Only the one for which $|\theta_1|<1$ is invertible and relevant.
