Thanks to Kit Baum, I have made available on SSC a new post estimation command -suregr- (mnemonic for -suregr[obust]-) that works after -[R] sureg -- Zellner's seemingly unrelated regression- and produces robust, or cluster-robust variance for the Seemingly Unrelated Regressions model.

You can install -suregr- from within Stata by typing

Code:
 ssc describe suregr

 ssc install suregr
The motivation for writing this command is that Stata's native -sureg- is not able to calculate (cluster-)robust variance and standard errors for the parameter estimates, and this remains true as of Stata 17 which came out today.

The command -suregr- implements automatically the methods and formulas in my recent paper Kolev (2021) "Robust and/or cluster-robust variance estimation in Seemingly Unrelated Regressions, and in Multivariate Regressions," and if you are interested in the methods and formulas I can send you the paper.

What my -robustr- does is replace the standard variance with the (cluster-) robust variance and display -sureg- table with the robust standard errors. Everything that works with -sureg- still works as before, say you can impose linear constraints in the -sureg- estimation stage, and you can use -test-, -nlcom- and -lincom- after -suregr- as usual.

Examples of use follow below:

Code:
. sysuse auto, clear
(1978 Automobile Data)

. 
. * Typical use of -suregr-, we firstly fit -sureg- quitetly, and then we follow up with -sureg- 
. * to get the robust variance.
. * If we do not specify the minus option, the default is minus(0) meaning no degrees of freedom adjustmen
> t.
. * Of course if we want to see the non-robust standard errors as well we can also run -sureg- noisily.  
. 
. quietly sureg (price headroom) (mpg weight length)

. 
. suregr

Seemingly unrelated regression
--------------------------------------------------------------------------
Equation             Obs   Parms        RMSE    "R-sq"       chi2        P
--------------------------------------------------------------------------
price                 74       1    2910.242    0.0131       0.94   0.3314
mpg                   74       2     3.34875    0.6604     136.53   0.0000
--------------------------------------------------------------------------

------------------------------------------------------------------------------
             |               Robust
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
price        |
    headroom |   389.6366   295.9122     1.32   0.188    -190.3406    969.6139
       _cons |   4998.979   843.8949     5.92   0.000     3344.976    6652.983
-------------+----------------------------------------------------------------
mpg          |
      weight |  -.0033167   .0019688    -1.68   0.092    -.0071756    .0005422
      length |  -.0916353   .0682694    -1.34   0.180    -.2254408    .0421703
       _cons |   48.53323   7.544959     6.43   0.000     33.74538    63.32108
------------------------------------------------------------------------------

. 
. * Typical use for getting the cluster-robust variance. We use the minus(1) degree of freedom adjustment,
>  
. * because this is the common adjustment cluster-robust variance estimators in native Stata commands use.
. * We also use the option noheader to omit the header from our regression table. 
. 
. quietly sureg (price headroom) (mpg weight length)

. 
. suregr, cluster(rep) minus(1) noheader

                                  (Std. Err. adjusted for 5 clusters in rep78)
------------------------------------------------------------------------------
             |               Robust
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
price        |
    headroom |    382.429   119.2846     3.21   0.001     148.6355    616.2224
       _cons |   4998.757   383.9062    13.02   0.000     4246.314    5751.199
-------------+----------------------------------------------------------------
mpg          |
      weight |  -.0027392   .0012197    -2.25   0.025    -.0051298   -.0003487
      length |  -.1121001   .0352499    -3.18   0.001    -.1811887   -.0430115
       _cons |   50.70261   6.473076     7.83   0.000     38.01561     63.3896
------------------------------------------------------------------------------

. 
. 
. * Post estimation after -suregr- works as usual.
. * -suregr- has substituted the (cluster-)robust variance, and the postestimation commands like -test-
. * -lincom- and -nlcom- use  the (cluster-)robust variance in the calculations they are carrying out:
. 
. quietly sureg (price headroom) (mpg weight length)

. 
. suregr, cluster(rep) minus(1) noheader

                                  (Std. Err. adjusted for 5 clusters in rep78)
------------------------------------------------------------------------------
             |               Robust
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
price        |
    headroom |    382.429   119.2846     3.21   0.001     148.6355    616.2224
       _cons |   4998.757   383.9062    13.02   0.000     4246.314    5751.199
-------------+----------------------------------------------------------------
mpg          |
      weight |  -.0027392   .0012197    -2.25   0.025    -.0051298   -.0003487
      length |  -.1121001   .0352499    -3.18   0.001    -.1811887   -.0430115
       _cons |   50.70261   6.473076     7.83   0.000     38.01561     63.3896
------------------------------------------------------------------------------

. 
. test [price]headroom = [mpg]weight

 ( 1)  [price]headroom - [mpg]weight = 0

           chi2(  1) =   10.28
         Prob > chi2 =    0.0013

. 
. nlcom [price]headroom/[mpg]weight

       _nl_1:  [price]headroom/[mpg]weight

------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _nl_1 |  -139611.7   70247.72    -1.99   0.047    -277294.7   -1928.649
------------------------------------------------------------------------------

. 
. * Imposing linear constraints works as usual too
. 
. constraint define 1 [price]headroom = [mpg]weight

. 
. quietly sureg (price headroom) (mpg weight length), constraint(1)

. 
. suregr,  noheader

 ( 1)  [price]headroom - [mpg]weight = 0
------------------------------------------------------------------------------
             |               Robust
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
price        |
    headroom |  -.0025949   .0020145    -1.29   0.198    -.0065432    .0013534
       _cons |   6165.265   340.5458    18.10   0.000     5497.807    6832.722
-------------+----------------------------------------------------------------
mpg          |
      weight |  -.0025949   .0020145    -1.29   0.198    -.0065432    .0013534
      length |  -.1057886   .0695312    -1.52   0.128    -.2420672      .03049
       _cons |   49.01368   7.647086     6.41   0.000     34.02566    64.00169
------------------------------------------------------------------------------

. 
. 
.
Reference

Kolev, Gueorgui I. (2021). Robust and/or cluster-robust variance estimation in Seemingly Unrelated Regressions, and in Multivariate Regressions.
(under review at the Stata Journal, available upon request from the author).