Entity fixed effects vs. time fixed effects - opposite results

#1
I'm analyzing some panel data that show how the percentage of the workforce as well as median wages have changed over time (hryear4) across major industry groupings (prmjind1). The data look like this:

Code:
Row │ hryear4  prmjind1  prcnt_minority  median_wage  wage_10  median_age
     │ Int64    Int64     Float64         Float64      Float64  Float64   
─────┼─────────────────────────────────────────────────────────────────────
   1 │    2010         1        54.9525      10.0      8.0           36.0
   2 │    2010         2        28.5844      18.0     11.0242        40.0
   3 │    2010         3        38.9158      16.0     10.0           37.0
   4 │    2010         4        37.2013      14.5      9.0           43.0
   5 │    2010         5        33.117       10.0      7.5           34.0
If I do a simple OLS regression where median_wage = a + b * percent_minority, I get the following output:
Code:
Continuous Response Model
Number of observations: 143
Null Loglikelihood: -366.63
Loglikelihood: -353.41
R-squared: 0.1689
LR Test: 26.43 ∼ χ²(1) ⟹  Pr > χ² = 0.0000
Formula: median_wage ~ 1 + prcnt_minority
Variance Covariance Estimator: OIM
──────────────────────────────────────────────────────────────────────────────
                    PE         SE      t-value  Pr > |t|      2.50%     97.50%
──────────────────────────────────────────────────────────────────────────────
(Intercept)     23.5535    1.62157    14.5251     <1e-29  20.3478    26.7593
prcnt_minority  -0.211443  0.0395056  -5.35222    <1e-06  -0.289543  -0.133343
This model says that for every 1 unit increase in the percent of the workforce that are minorities, you get a 21 cent drop in the median wage. Now, if I do a fixed-effect regression assuming that there exist unobserved time-invariant differences across industries, the results change completely:

Code:
Continuous Response Model
Number of observations: 143
Null Loglikelihood: -366.63
Loglikelihood: -180.77
R-squared: 0.9261
Wald: 162.53 ∼ F(1, 129) ⟹ Pr > F = 0.0000
Formula: median_wage ~ 1 + prcnt_minority + absorb(prmjind1)
Variance Covariance Estimator: OIM
──────────────────────────────────────────────────────────────────────────────
                    PE         SE        t-value  Pr > |t|     2.50%    97.50%
──────────────────────────────────────────────────────────────────────────────
(Intercept)     0.0584638  1.17216     0.0498769    0.9603  -2.26068  2.37761
prcnt_minority  0.367398   0.0288186  12.7486       <1e-23   0.31038  0.424417
──────────────────────────────────────────────────────────────────────────────
This model says that every 1 unit increase in the percent of the workforce that are minorities results in a 36 cent increase in the median wage. Lastly, if I do another fixed-effect regression but this time assume that there exist constant differences across industries that vary over time, the results again change dramatically:

Code:
Continuous Response Model
Number of observations: 143
Null Loglikelihood: -366.63
Loglikelihood: -317.78
R-squared: 0.4969
Wald: 87.84 ∼ F(1, 131) ⟹ Pr > F = 0.0000
Formula: median_wage ~ 1 + prcnt_minority + absorb(hryear4)
Variance Covariance Estimator: OIM
──────────────────────────────────────────────────────────────────────────────
                    PE         SE      t-value  Pr > |t|      2.50%     97.50%
──────────────────────────────────────────────────────────────────────────────
(Intercept)     27.9752    1.40108    19.9668     <1e-40  25.2035    30.7469
prcnt_minority  -0.320378  0.0341831  -9.37241    <1e-15  -0.388001  -0.252756
──────────────────────────────────────────────────────────────────────────────
Now we're back to a decrease in the median wage, and a much bigger decrease than the one estimated from OLS. I'm looking for advice regarding how to interpret these findings and what next steps would be. If the coefficient always had the same sign, it would be much easier but I don't understand how it can flip from positive to significantly negative like it's doing here.